How smooth can the convex hull of a Lévy path be?

Abstract

We describe the rate of growth of the derivative C′C' of the convex minorant of a L\'evy path on a set of times, recently characterised in our preceding paper, where C′C' increases continuously. Since the convex minorant is piecewise linear, C′C' may exhibit such behaviour either at the vertex time τs\tau_s of slope s or at time 0. The analysis at times 0 and τs\tau_s each require different tools, but an interesting diagonal connection arises: the structure of the upper (resp. lower) functions at time 0 is similar to that of the lower (resp. upper) functions at time τs\tau_s. Even though the convex hull depends on the entire path, we show that the fluctuations of the derivative C′C' depend only on the fine structure of the small jumps of the L\'evy process and do not depend on the time horizon. For L\'evy processes attracted to a stable process, we establish sharp results essentially characterising the modulus of continuity of the derivative of the boundary of the convex hull at such times up to sublogarithmic factors. As a consequence of the relation between the lower fluctuations of the convex minorant and the path of the L\'evy process, we obtain novel results for the growth rate at 0 for the meanders of a wide class of L\'evy processes.

Full text

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE?
DAVID BANG, JORGE GONZ´
ALEZ C ´
AZARES & ALEKSANDAR MIJATOVI´
C
Abstract. We describe the rate of growth of the derivative C0of the convex minorant of a L´evy
path on a set of times, recently characterised in [2], where C0increases continuously. Since the convex
minorant is piecewise linear, C0may exhibit such behaviour either at the vertex time τsof slope s
or at time 0. The analysis at times 0 and τseach require different tools, but an interesting diagonal
connection arises: the structure of the upper (resp. lower) functions at time 0 is similar to that of the
lower (resp. upper) functions at time τs. Even though the convex hull depends on the entire path,
we show that the fluctuations of the derivative C0depend only on the fine structure of the small
jumps of the L´evy process and do not depend on the time horizon. For L´evy processes attracted
to a stable process, we establish sharp results essentially characterising the modulus of continuity of
the derivative of the boundary of the convex hull at such times up to sublogarithmic factors. As a
consequence of the relation between the lower fluctuations of the convex minorant and the path of
the L´evy process, we obtain novel results for the growth rate at 0 for the meanders of a wide class of
L´evy processes.
1. Introduction
The class of L´evy processes with paths whose graphs have convex hulls in the plane with smooth
boundary almost surely has recently been characterised in [2]. In fact, as explained in [2], to understand
whether the boundary is smooth at a point with tangent of a given slope, it suffices to analyse whether
the right-derivative C0= (C0
t)t∈(0,T )of the convex minorant C= (Ct)t∈[0,T ]of a L´evy process X=
(Xt)t∈[0,T ]is continuous as it attains that slope (recall that Cis the pointwise largest convex function
satisfying Ct≤Xtfor all t∈[0, T ]). The main objective of this paper is to quantify the smoothness of
the boundary of the convex hull of Xby quantifying the modulus of continuity of C0via its lower and
upper functions. In the case of times 0 and T, we quantify the degree of smoothness of the boundary of
the convex hull by analysing the rate at which |C0
t| → ∞ as tapproaches either 0 or T(see YouTube [3]
for a short presentation of our results).
It is known that Cis a piecewise linear convex function [17,29] and the image of the right-derivative
C0over the open intervals of linearity of Cis a countable random set Swith a.s. deterministic limit
points that do not depend on the time horizon T, see [2, Thm 1.1]. These limit points of Sdetermine
the continuity of C0on (0, T ) outside of the open intervals of constancy of C0, see [2, App. A]. Indeed,
the vertex time process τ= (τs)s∈R, given by τs:= inf{t∈(0, T ) : C0
t> s}∧T(where a∧b:= min{a, b}
and inf ∅:=∞), is the right-inverse of the non-decreasing process C0. The process τfinds the times in
[0, T ] of the vertices of the convex minorant C(see [17, Sec. 2.3]), so the only possible discontinuities
of C0lie in the range of τ. Clearly, it suffices to analyse only the times τsfor which C0is non-constant
on the interval [τs, τs+ε) for every ε > 0 (otherwise, τsis the time of a vertex isolated from the right).
At such a time, the continuity of C0can be described in terms of a limit set of S. In the present paper
we analyse the quality of the right-continuity of C0at such points. By time reversal, analogous results
Date: June 20, 2022.
2020 Mathematics Subject Classification. 60G51,60F15.
Key words and phrases. Derivative of convex minorant, L´evy processes, law of iterated logarithm, additive processes.
1

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 2
apply for the left-continuity of t7→ C0
ton (0, T ) (i.e., as t↑τsfor s∈R) and for the explosion of C0
t
as t↑T. Throughout the paper, the variable s∈Rwill be reserved for slope, indexing the vertex time
process τ.
1.1. Contributions. We describe the small-time fluctuations of the derivative of the boundary of the
convex hull of Xat its points of smoothness. This requires studying the local growth of C0in two
regimes: at finite slope (FS) sin the deterministic set L+(S)⊂Rof right-limit points1of the set
of slopes Sand at infinite slope (IS) for L´evy processes of infinite variation, see Figure 1below. In
terms of times, regime (FS) with s∈ L+(S) analyses how C0leaves the slope sat vertex time τsin
[0, T ) and regime (IS) analyses how C0enters from −∞ at time 0 = limu↓−∞ τu. At all other times
t∈(0, T )\ {τs:s∈ L+(S)}, the derivative C0is constant on [t, t +ε) for some sufficiently small
ε > 0. In particular, in what follows we exclude all L´evy processes that are compound Poisson with
drift, since C0only takes finitely many values in that case.
0.0 0.1 0.2 0.3 0.4 0.5 0.6
t
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
Lévy path Xt
Convex minorant Ct
0.0 0.1 0.2 0.3 0.4 0.5
t
0.00
0.05
0.10
0.15
Post−τ0 Lévy path Xt+τ0−Xτ0
Post−τ0 convex minorant Ct+τ0−Cτ0
Figure 1. The picture on the left shows the path of an α-stable L´evy process Xwith
α∈(1,2) and its convex minorant Cstarting at time 0. The picture on the right shows the
post-minimum process (Xt+τ0−Xτ0)t∈[0,T −τ0]of an α-stable process with α∈(0,1) and its
corresponding convex minorant (Ct+τ0−Cτ0)t∈[0,T−τ0]. Note that, in the case α∈(0,1),
the derivative C0is continuous only at τ0, i.e. at t= 0 in the graph, and at no other contact
point between the path and its convex minorant.
Regime (FS): C0immediately after τs.Given a slope s∈R, we have s /∈ S a.s. by [17,
Thm 3.1] since the law of Xis diffuse. By [2, Thm 1.1], s∈ L+(S) if and only if the derivative
C0attains level sat a unique time τs∈(0, T ) (i.e. C0
τs=s) and is not constant on every interval
[τs, τs+ε), ε > 0, a.s. Moreover, s∈ L+(S) if and only if R1
0P(Xt/t ∈(s, s +ε))t−1dt=∞for all
ε > 0. The regime (FS) includes an infinite variation process Xif it is strongly eroded (implying
L+(S) = R) or, more generally, if (Xt−st)t≥0is eroded (implying s∈ L+(S)), see [2]. Moreover,
regime (FS) includes a finite variation process Xat slope s∈ L+(S) if and only if the natural drift
γ0= limt↓0Xt/t equals sand R1
0P(Xt> γ0t)t−1dt=∞or, equivalently, if the positive half-line is
regular for (Xt−γ0t)t≥0(see [2, Cor. 1.4] for a characterisation in terms of the L´evy measure of Xor
its characteristic exponent).
Our results in regime (FS) are summarised as follows. For any process with s∈ L+(S), Theorem 2.2
establishes general sufficient conditions identifying when liminft↓0(C0
t+τs−s)/f(t) is either 0 a.s. or
∞a.s. In particular, we show that lim inf t↓0(C0
t+τs−s)/f(t) cannot take a positive finite value if X
1A point xis a right-limit point of A⊂R, denoted x∈ L+(A) if A∩(x, x +ε)6=∅for all ε > 0 (see also [2, App. A]).

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 3
has jumps of both signs and is an α-stable with α∈(0,1] (recall that, if α > 1, then L+(S) = ∅by [2,
Prop. 1.6]).
For processes Xin the small-time domain of attraction of an α-stable process with α∈(0,1) (see
Subsection 2.2 below for definition), Theorem 2.7 finds a parametric family of functions fthat es-
sentially determine the upper fluctuations of C0
t+τs−sup to sublogarithmic factors. In particular,
Theorem 2.7 determines when lim supt↓0(C0
t+τs−s)/f(t) equals 0 a.s. or ∞a.s., essentially character-
ising the right-modulus of continuity2of C0at τs. The family of functions fis given in terms of the
regularly varying normalising function of X.
Regime (IS): C0immediately after 0.The boundary of the convex hull of Xis smooth at the
origin if and only if limt↓0C0
t=−∞ a.s., which is equivalent to Xbeing of infinite variation (see [2,
Prop. 1.5 & Sec. 1.1.2]). If Xhas finite variation, then C0is bounded (see [2, Prop. 1.3]). In this case,
C0has positive probability of being non-constant on the interval [0, ε) for every ε > 0 if and only if the
negative half-line is not regular. Moreover, if this event occurs, then C0
tapproaches the natural drift
γ0as t↓0 by [2, Prop. 1.3(b)] and the local behaviour of C0at 0 would be described by the results of
regime (FS). Thus, in regime (IS) we only consider L´evy processes of infinite variation.
Our results in regime (IS) are summarised as follows. For any infinite variation process X, Theo-
rem 2.9 establishes general sufficient conditions for lim supt↓0|C0
t|f(t) to equal either 0 a.s. or ∞a.s.
In particular, we show that limsupt↓0|C0
t|f(t) cannot take a positive finite value if Xis α-stable with
α∈[1,2) and has (at least some) negative jumps.
If the L´evy process lies in the domain of attraction of an α-stable process, with α∈(1,2], Theo-
rem 2.13 finds a parametric family of functions fthat essentially determine the lower fluctuations of C0
up to sublogarithmic functions. The function fis given in terms of the regularly varying normalising
function of X. Again, these results describe the right-modulus of continuity of the derivative of the
boundary of the convex hull of X(as a closed curve in R2) at the origin. In this case, for a sufficiently
small ε > 0, we may locally parametrise the curve ((t, Ct); t∈[0, ε]), as ((ς(t), t); t∈[Cε,0]), using a
local inverse ς(t) of Ctwith left-derivative ς0(t) = 1/C0
ς(t)that vanishes at 0 (since limt↓01/|C0
t|= 0
a.s.). Thus, the left-modulus of continuity of ςat 0 is described by the upper and lower limits of
(|C0
t|f(t))−1as t↓0, the main focus of our results in this regime.
Consequences for the path of a L´evy process and its meander. In Subsection 2.5 we
present some implications the results in this paper have for the path of X. We find that, under certain
conditions, the local fluctuations of Xcan be described in terms of those of C0, yielding novel results
for the local growth of the post-minimum process of Xand the corresponding L´evy meander (see
Lemma 2.15 and Corollaries 2.16 and 2.17 below).
1.2. Strategy and ideas behind the proofs. An overview of the proofs of our results is as follows.
First we show that, under our assumptions, the local properties of C0do not depend on the time
horizon T. This reduces the problem to the case where the time horizon Tis independent of Xand
exponentially distributed (the corresponding right-derivative is denoted b
C0). Second, we translate
the problem of studying the local behaviour of b
C0to the problem of studying the local behaviour
of its inverse: the vertex time process bτ. Third, we exploit the fact that, since the time horizon T
is an independent exponential random variable with mean 1/λ, the vertex time process bτis a time-
inhomogenous non-decreasing additive process (i.e., a process with independent but non-stationary
2We say that a non-decreasing function ϕ: [0,∞)→[0,∞) is a right-modulus of continuity of a right-continuous
function gat x∈Rif lim supy↓x|g(y)−g(x)|/ϕ(y−x)<∞.

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 4
increments) and its Laplace exponent is given by (see [17, Thm 2.9]):
(1) E[e−wbτu] = e−Φu(w),where Φu(w):=Z∞
0
(1 −e−wt)e−λt P(Xt≤ut)dt
t,for w≥0, u∈R.
These three observations reduce the problem to the analysis of the fluctuations of the additive process bτ.
The local properties of C0are entirely driven by the small jumps of X. However, different facets
of the small-jump activity of Xdominate in each regime, resulting in related but distinct results and
criteria. Indeed, regime (FS) corresponds to the short-term behaviour of bτs+u−bτsas u↓0 while
regime (IS) corresponds to the long-term behaviour of bτuas u→ −∞ (note that, when Xis of infinite
variation, τu>0 for u∈Rand limu→−∞ bτu= 0 a.s.). This bears out in a difference in the behaviour
of the Laplace exponent Φ of bτat either bounded or unbounded slopes and leads to an interesting
diagonal connection in behaviour that we now explain.
Our main tool is the novel description of the upper and lower fluctuations of a non-decreasing time-
inhomogenous additive process Ystarted at Y0= 0, in terms of its time-dependent L´evy measure and
Laplace exponent. In our applications, the process Yis given by (bτu+s−bτs)u≥0in regime (FS) and
(bτ−1/u)u≥0(with conventions −1/0 = −∞ and bτ−∞ = 0) in regime (IS). Then our main technical
tools, Theorems 3.1 and 3.3 of Section 3below, describing the upper and lower fluctuations of Y, also
serve to describe the lower and upper fluctuations, respectively, of the right-inverse Lof Y. Since,
in regime (FS), we have b
C0
t+τs−s=Ltbut, in regime (IS), we have b
C0
t=−1/Lt, the lower (resp.
upper) fluctuations of b
C0in regime (FS) will have a similar structure to the upper (resp. lower)
fluctuations of b
C0in regime (IS). This diagonal connection is a priori surprising as the processes
considered by either regime need not have a clear connection to each other. Indeed, regime (FS)
considers most finite variation processes and only some infinite variation processes while regime (IS)
considers exclusively infinite variation processes. This diagonal connection is reminiscent of the duality
between stable process with stability index α∈(1,2] and a corresponding stable process with stability
index 1/α ∈[1/2,1) arising in the famous time-space inversion first observed by Zolotarev for the
marginals and later studied by Fourati [14] for the ascending ladder process (see also [21] for further
extensions of this duality).
The lower and upper fluctuations of the corresponding process Yrequire varying degrees of control
on its Laplace exponent Φ in (1). The assumptions of Theorem 3.1 require tight two-sided estimates
of Φ, not needed in Theorem 3.3. When applying Theorem 3.1, we are compelled to assume Xlies in
the domain of attraction of an α-stable process. In regime (FS) this assumption yields sharp estimates
on the density of Xtas t↓0, which in turn allows us to control the term P(0 < Xt−st ≤ut) for
small t > 0 in the Laplace exponent Φs+u−Φsof bτu+s−bτsas u↓0, cf. (1) above. The growth rate
of the density of Xtas t↓0 is controlled is by lower estimates on the small-jump activity of Xgiven
in Lemma 4.4 below, a refinement of the results in [28] for processes attracted to a stable process. In
regime (IS) we require control over the negative tail probabilities P(Xt≤ut) for small t > 0 appearing
in the Laplace exponent Φuof bτuas u→ −∞, cf. (1). The behaviour of these tails are controlled by
upper estimates of the small-jump activity of X, which are generally easier to obtain. In this case,
moment bounds for the small-jump component of the L´evy process and the convergence in Kolmogorov
distance implied by the attraction to the stable law, give sufficient control over these tail probabilities.
1.3. Connections with the literature. In [8], Bertoin finds the law of the convex minorant of
Cauchy process on [0,1] and finds the exact asymptotic behaviour (in the form of a law of interated
logarithm with a positive finite limit) for the derivative C0at times 0, 1 and any τs,s∈R. The
methods in [8] are specific to Cauchy process with its linear scaling property, making the approach
hard to generalise. In fact, the results in [8] are a direct consequence of the fact that the vertex time

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 5
process bτhas a Laplace transform Φ in (1) that factorises as Φu(w) = P(X1≤u)Φ∞(w), making bτa
gamma subordinator under the deterministic time-change u7→ P(X1≤u), cf. Example 4.3 below.
Paul L´evy showed that the boundary of the convex hull of a planar Brownian motion has no corners
at any point, see [24], motivating [13] to characterise the modulus of continuity of the derivative of that
boundary. Given the recent characterisation of the smoothness of the convex hull of a L´evy path [2],
the results in the present paper are likewise motivated by the study of the modulus of continuity of
the derivative of the boundary in this context.
The literature on the growth rate of the path of a L´evy process Xis vast, particularly for subordina-
tors, see e.g. [7,15,16,22,32,33,37]. The authors in [15,16] study the growth rate of a subordinator
at 0 and ∞. In [15] (see also [7, Prop 4.4]) Fristedt fully characterises the upper fluctuations of a
subordinator in terms of its L´evy measure, a result we generalise in Theorem 3.3 to processes that
need not have stationary increments. In [7, Thm 4.1] (see also [16, Thm 1], a function essentially
characterising the exact lower fluctuations of a subordinator is constructed in terms of its Laplace
exponent. These methods are not easily generalised to the time-inhomogenous case since the Laplace
exponent is now bivariate and there is neither a one-parameter lower function to propose nor a clear
extension to the proofs.
In [31], Sato establishes results for time-inhomogeneous non-decreasing additive processes similar
to our result in Section 3. The assumptions in [31] are given in terms of the transition probabilities
of the additive process, which are generally intractable, particularly for the processes (bτ−1/u)u>0and
(bτu+s−bτs)u≥0, considered here. Our results are also easier to apply in other situations as well, for
example, to fractional Poisson processes (see definition in [5]).
The upper fluctuations of a L´evy process at zero have been the topic of numerous studies, see [6,
32] for the one-sided problem and [22,33,37] for the two-sided problem. Similar questions have
been considered for more general time-homogeneous Markov processes [12,23]. The time-homogeneity
again plays an important role in these results. The lower fluctuations of a stochastic process is only
qualitatively different from the upper fluctuations if the process is positive. This is the reason why
this problem has mostly only been addressed for subordinators (see the references above) and for the
running supremum of a L´evy process, see e.g. [1]. We stress that the results in the present paper,
while related in spirit to this literature, are fundamentally different in two ways. First, we study the
derivative of the convex minorant of a L´evy path on [0, T ], which (unlike e.g. the running supremum)
cannot be constructed locally from the restriction of the path of the L´evy process to any short interval.
Second, the convex minorant and its derivative are neither Markovian nor time-homogeneous. In fact,
the only result in our context prior to our work is in the Cauchy case [8], where the derivative of
the convex minorant is an explicit gamma process under a deterministic time-change, cf. Example 4.3
below.
1.4. Organisation of the article. In Section 2we present the main results of this article. We split
the section in four, according to regimes (FS) and (IS) and whether the upper or lower fluctuations of
C0are being described. The implications of the results in Section 2for the L´evy process and meander
are covered in Subsection 2.5. In Section 3, technical results for general time-inhomogeneous non-
decreasing additive processes are established. Section 4recalls from [17] the definition and law of the
vertex time process τand provides the proofs of the results stated in Section 2. Section 5concludes
the paper.

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 6
2. Growth rate of the derivative of the convex minorant
Let X= (Xt)t≥0be an infinite activity L´evy process (see [30, Def. 1.6, Ch. 1]). Let C= (Ct)t∈[0,T ]
be the convex minorant of Xon [0, T ] for some T > 0. Put differently, Cis the largest convex function
that is piecewise smaller than the path of X(see [17, Sec. 3, p. 8]). In this section we analyse the growth
rate of the right derivative of C, denoted by C0= (C0
t)t∈(0,T ), near time 0 and at the vertex time τs=
inf{t > 0 : C0
t> s}∧Tof the slope s∈R(i.e., the first time C0attains slope s). More specifically, we
give sufficient conditions to identify the values of the possibly infinite limits (for appropriate increasing
functions fwith f(0) = 0): lim supt↓0(C0
t+τs−s)/f(t) & lim inft↓0(C0
t+τs−s)/f(t) in the finite slope
(FS) regime and lim supt↓0|C0
t|f(t) & lim inf t↓0|C0
t|f(t) in the infinite slope (IS) regime. The values of
these limits are constants in [0,∞] a.s. by Corollary 4.2 below. We note that these limits are invariant
under certain modifications of the law of X, which we describe in the following remark.
Remark 2.1.
(a) Let Pbe the probability measure on the space where Xis defined. If the limits lim supt↓0|C0
t|f(t),
lim inft↓0|C0
t|f(t), lim supt↓0(C0
t+τs−s)/f(t) and lim inf t↓0(C0
t+τs−s)/f(t) are P-a.s. constant,
then they are also P0-a.s. constant with the same value for any probability measure P0absolutely
continuous with respect to P. In particular, we may modify the L´evy measure of Xon the
complement of any neighborhood of 0 without affecting these limits (see e.g. [30, Thm 33.1–33.2]).
(b) We may add a drift process to Xwithout affecting the limits at 0 since such a drift would only
shift |C0
t|by a constant value and f(t)→0 as t↓0. Similarly, for the limits of (C0
t+τs−s)/f(t)
as t↓0, it suffices to analyse the post-minimum process (i.e., the vertex time τ0) of the process
(Xt−st)t≥0. For ease of reference, our results are stated for a general slope s.♦
2.1. Regime (FS): lower functions at time τs.The following theorem describes the lower fluctu-
ations of C0
t+τs−sas t↓0. Recall that L+(S) is the a.s. deterministic set of right-limit points of the
set of slopes S.
Theorem 2.2. Let s∈ L+(S)and fbe continuous and increasing, satisfying f(t)≤1 = f(1) for
t∈(0,1] and f(0) = 0 = limc↓0lim supt↓0f(ct)/f(t). Let c > 0and consider the following conditions:
Z1
0
P(0 <(Xt−st)/t ≤f(t/c))dt
t<∞,(2)
Z1
0
Et
f−1((Xt−st)/t)21{f(t/2)<(Xt−st)/t≤1}dt < ∞,(3)
2nZ2−n
0
P(f(t/2) <(Xt−st)/t ≤f(2−n))dt→0,as n→ ∞.(4)
Then the following statements hold.
(i) If (2)–(4)hold for c= 1, then lim inft↓0(C0
t+τs−s)/f(t) = ∞a.s.
(ii) If (2)fails for every c > 0, then lim inft↓0(C0
t+τs−s)/f(t)=0a.s.
(iii) If lim inft↓0(C0
t+τs−s)/f(t)>1a.s., then (2)holds for any c > 1.
Some remarks are in order.
Remark 2.3.
(a) Any continuous regularly varying function fof index r > 0 satisfies the assumption in the theorem:
limc↓0limt↓0f(ct)/f(t) = limc↓0cr= 0. Moreover, the assumption f(t)≤1 = f(1) for t∈(0,1] is
not necessary but makes conditions (2)–(4) take a simpler form.

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 7
(b) The proof of Theorem 2.2 is based on the analysis of the upper fluctuations of τat slope s.
Condition (2) ensures (τu+s−τs)u≥0jumps finitely many times over the boundary u7→ f−1(u),
condition (4) makes the small-jump component of (τu+s−τs)u≥0(i.e. the sum of the jumps at
times v∈[s, u +s] of size at most f−1(v)) have a mean that tends to 0 as u↓0 and condition (3)
controls the deviations of (τu+s−τs)u≥0away from its mean.
(c) Note that (4) holds if R1
0P(f(2−nt/2) <(X2−nt−s2−nt)/(2−nt)≤f(2−n))dt→0 as n→ ∞,
which, by the dominated convergence theorem, holds if P(f(u/2) <(Xu−su)/u ≤f(u/t)) →0
as u↓0 for a.e. t∈(0,1).
(d) Condition (3) in Theorem 2.2 requires access to the inverse f−1of the function f. In the special
case when the function fis concave, this assumption can be replaced with an assumption given
in terms of f(cf. Proposition 3.5 and Corollary 3.7). However, it is important to consider
non-concave functions f, see Corollary 2.4 below. ♦
2.1.1. Simple sufficient conditions for the assumptions of Theorem 2.2.Let fbe as in Theorem 2.2.
By Theorem 3.3(c) below (with the measure Π(dx, dt) = P((Xt−st)/t ∈dx)t−1dt), the following
condition implies (3)–(4):
(5) Z1
0
E1
f−1((Xt−st)/t)1{f(t/2)<(Xt−st)/t≤1}dt < ∞.
If estimates on the density of Xtare available (e.g., via assumptions on the generating triplet of X),
(5) can be simplified further, see Corollary 2.4 below.
Throughout, we denote by (γ, σ2, ν ) the generating triplet of X(corresponding to the cutoff function
x7→ 1(−1,1)(x), see [30, Def. 8.2]), where γ∈Ris the drift parameter, σ2≥0 is the Gaussian coefficient
and νis the L´evy measure of Xon R. We also define the functions
σ2(ε):=σ2+σ2
+(ε) + σ2
−(ε), σ2
+(ε):=Z(0,ε)
x2ν(dx), σ2
−(ε):=Z(−ε,0)
x2ν(dx),for ε > 0.
Recall that, in regime (FS), we have σ2= 0 (see [2, Prop. 1.6]). Given two positive functions g1and
g2, we say g1(ε) = O(g2(ε)) as ε↓0 if limsupε↓0g1(ε)/g2(ε)<∞. Similarly, we write g1(ε)≈g2(ε) as
ε↓0 if g1(ε) = O(g2(ε)) and g2(ε) = O(g1(ε)).
Corollary 2.4. Fix β∈(0,1] and let s∈ L+(S)and fbe as in Theorem 2.2.
(a) If lim infε↓0εβ−2σ2(ε)>0,fis differentiable with positive derivative f0>0and the integrals
R1
0R1
t/2(f0(y)/y)t1−1/β dydtand R1
0t−1/βf(t)dtare finite, then lim inft↓0(C0
t+τs−s)/f(t) = ∞a.s.
(b) Assume R1
0((t−1/βf(t)) ∧t−1)dt=∞and either of the following hold:
(i) σ2(ε)≈εand |R(−1,1)\(−ε,ε)xν(dx)|=O(1) as ε↓0,
(ii) β∈(0,1) and σ2
±(ε)≈ε2−βas ε↓0for both signs of ±,
then lim inft↓0(C0
t+τs−s)/f(t)=0a.s.
We stress that the sufficient conditions in Corollary 2.4 are all in terms of the characteristics of the
L´evy process Xand the function f.
Remark 2.5.
(a) The assumptions in Corollary 2.4 are satisfied by most processes in the class Zα,ρ of L´evy processes
in the small-time domain of attraction of an α-stable distribution, see Subsection 2.2 below (cf. [19,
Eq. (8)]). Thus, the assumptions of part (a) in Corollary 2.4 hold for any X∈ Zα,ρ and β < α
(by Karamata’s theorem [9, Thm 1.5.11], we can take β=αif the normalising function gof X
satisfies lim inf t↓0t−1/αg(t)>0). Moreover, the assumptions of cases (b-i) and (b-ii) hold for
processes in the domain of normal attraction (i.e. if the normalising function equals g(t) = t1/α

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 8
for all t > 0) with ρ∈(0,1) and β=α∈(0,1], see [19, Thm 2]. In particular, these assumptions
are satisfied by stable processes with α∈(0,1] and ρ∈(0,1).
(b) Both integrals in part (a) of Corollary 2.4 are finite or infinite simultaneously whenever f0is
regularly varying at 0 with nonzero index by Karamata’s theorem [9, Thm 1.5.11]. Thus, in that
case, under the conditions of either (b-i) or (b-ii), the limit lim inf t↓0(C0
t+τs−s)/f(t) equals 0 or
∞according to whether R1
0t−1/βf(t)dtis infinite or finite, respectively.
(c) The case β > 1 is not considered in Corollary 2.4(a) and (b-ii) since in this case we would have
L+(S) = ∅by [2, Prop. 1.6]. ♦
Proof of Corollary 2.4.Assume without loss of generality that s= 0 ∈ L+(S) (equivalently, we con-
sider the process (Xt−st)t≥0for s∈ L+(S)).
(a) Our assumptions and [28, Thm 3.1] show that the density x7→ pX(t, x) of Xtexists for t > 0
and moreover supx∈RpX(t, x)≤Ct−1/β for some C > 0 and all t∈(0,1]. Thus, (5) is implied by
(6) Z1
0Zt
tf(t/2)
1
f−1(x/t)t−1/βdxdt=Z1
0Z1
t/2
f0(y)
yt1−1/βdydt < ∞,
where we have used the change of variable x=tf(y). Similarly, the bound on the density of Xtshows
that condition (2) holds if R1
0t−1/βf(t)dt < ∞. Thus, the result follows from Theorem 2.2.
(b) In either case (i) or (ii), our assumptions and [28, Thm 4.3] show that Ct−1/β ≤pX(t, x) for
some C > 0 and all |x| ≤ t1/β . Thus P(0 < Xt≤tf(t/c)) ≥((tf (t/c)) ∧t1/β )Ct−1/β , implying that (2)
fails for some c > 0 whenever R1
0((t−1/βf(t/c)) ∧t−1)dt=∞. A simple change of variables shows
that this integral is either finite for all c > 0 or infinite for all c > 0. The result then follows from
Theorem 2.2(ii). 
The following is another simple corollary of Theorem 2.2. This result can also be established using
similar arguments to those used in [8, Cor. 3], see the discussion ensuing the proof of [8, Cor. 3].
Corollary 2.6. Let Xbe a Cauchy process, fbe as in Theorem 2.2 and pick s∈R. Then the limit
lim inft↓0(C0
t+τs−s)/f(t)equals 0(resp. ∞) a.s. if R1
0t−1f(t)dtis infinite (resp. finite).
Proof. Assume without loss of generality that s= 0. Then the law of Xt/t does not depend on t > 0
and hence the integral in (5) equals
Z1
0
E1{t/2<f−1(X1)≤1}
f−1(X1)dt=EZ1
0
1{t/2<f−1(X1)≤1}
f−1(X1)dt≤2P(X1∈(0,1]) <∞.
Moreover, condition (2) simplifies to R1
0P(0 < X1≤f(t/c))t−1dt < ∞, which is equivalent to the
integral R1
0t−1f(t/c)dtbeing finite since X1has a bounded density that is bounded away from zero on
[0,1]. The change of variables t0=t/c shows that this integral is either finite for all c > 0 or infinite
for all c > 0. Thus, Theorem 2.2 gives the result. 
2.2. Regime (FS): upper functions at time τs.The upper fluctuations of C0
t+τs−sare harder
to describe than the lower fluctuations studied in Subsection 2.1 above. The main reason for this is
that in Theorem 2.7 below the lim sup of C0at a vertex time τscan be expressed in terms of the
lim inf of the vertex time process τ, which requires strong two-sided control on the Laplace exponent
Φu+s(w)−Φs(w), defined in (1), of the variable τu+s−τsas w→ ∞ and u↓0. (In the proof of
Theorem 2.2, lim sup of the vertex time process τis needed, which is easier to control.) In turn, by (1),
this requires sharp two-sided estimates on the probability P(0 < Xt−st ≤ut) as a function of (u, t)
for small u, t > 0. In particular, it is important to have strong control on the density of Xtfor small

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 9
t > 0 on the “pizza slice” {(t, x) : s < x/t ≤u+s}as u↓0. We establish these estimates for the
processes in the domain of attraction of an α-stable process, leading to Theorem 2.7 below.
We denote by Zα,ρ the class of L´evy processes in the small-time domain of attraction of an α-stable
process with positivity parameter ρ∈[0,1] (see [19, Eq. (8)]). In the case α < 1, relevant in the
regime (FS) at slope sequal to the natural drift γ0, for each L´evy process X∈ Zα,ρ there exists a
normalising function gthat is regularly varying at 0 with index 1/α and an α-stable process (Zu)u∈[0,T ]
with ρ=P(Z1>0) ∈[0,1] such that the weak convergence ((Xut −γ0ut)/g(t))u∈[0,T ]
d
−→ (Zu)u∈[0,T]
holds as t↓0. Given X∈ Zα,ρ with normalising function g, we define G(t):=t/g(t) for t∈(0,∞).
Theorem 2.7. Suppose X∈ Zα,ρ for some α∈(0,1) and ρ∈(0,1]. Define f: (0,1) →(0,∞)
through f(t):= 1/G(tlogp(1/t)),t∈(0,1), for some p∈R. Then the following hold for s=γ0:
(i) if p > 1/ρ, then lim supt↓0(C0
t+τs−s)/f(t)=0a.s.,
(ii) if p < 1/ρ, then lim supt↓0(C0
t+τs−s)/f(t) = ∞a.s.
The class Zα,ρ is quite large and the assumption X∈ Zα,ρ is essentially reduced to the L´evy measure
of Xbeing regularly varying at 0, see [19,§4] for a full characterisation of this class. In particular,
αagrees with the Blumenthal–Getoor index βBG defined in (13) below. Moreover, for α < 1 and
ρ∈(0,1], the assumption X∈ Zα,ρ implies that Xis of finite variation with P(Xt−γ0t > 0) →ρas
t↓0, implying L+(S) = {γ0}by [2, Prop. 1.3 & Cor. 1.4].
Note that the function fin Theorem 2.7 is regularly varying at 0 with index 1/α−1. The appearance
of the positivity parameter ρ, a nontrivial function of the L´evy measure of X, in Theorem 2.7 suggests
that the upper fluctuations of C0at time τs(for s=γ0) are more delicate than its lower fluctuations
described in Theorem 2.13. Indeed, if X∈ Zα,ρ is in the domain of normal attraction (i.e. g(t) = t1/α)
and ρ∈(0,1), then the fluctuations of C0at vertex time τs, characterised by Corollary 2.4(a) & (b-ii)
(with β=α) and Remark 2.5(a), do not involve parameter ρ. In particular, by Theorem 2.7 and
Corollary 2.4(b-ii), we have lim inft↓0(C0
t+τs−s)/f(t) = 0 and lim supt↓0(C0
t+τs−s)/f(t) = ∞a.s. for
f(t) = t1/α−1logq(1/t) and any q∈[−1,(1/α −1)/ρ), demonstrating the gap between the lower and
upper fluctuations of C0at vertex time τs.
Remark 2.8.
(a) The case where Xis attracted to Cauchy process with α= 1 is expected to hold for the functions
fin Theorem 2.7. For such X∈ Z1,ρ, a multitude of cases arise including Xhaving (i) less
activity (e.g., Xis of finite variation), (ii) similar amount of activity (i.e., Xis in the domain of
normal attraction) or (iii) more activity than Cauchy process (see, e.g. [2, Ex. 2.1–2.2]). In terms
of the normalising function gof X, these cases correspond to the limit limt↓0t−1/α g(t) being equal
to: (i) zero, (ii) a finite and positive constant or (iii) infinity. (Recall that in cases (ii) and (iii)
Xis strongly eroded with L+(S) = R, see [2, Ex. 2.1–2.2], and in case (i) Xmay be strongly
eroded, by [2, Thm 1.8], or of finite variation with L+(S) = {γ0}by [2, Prop 138] and the fact
that limt↓0P(Xt>0) = ρ∈(0,1).) However, we stress that our methodology can be used to
obtain a description of the lower fluctuations of C0at τsin cases (i), (ii) and (iii). This would
require an application of Theorem 3.1 along with two-sided estimates of the Laplace exponent Φ
of the vertex time process in (1), generalising Lemma 4.5 to the case α= 1. In the interest of
brevity we do not give the details of this extension.
(b) The boundary case p= 1/ρ can be analysed along similar lines. In fact, our methods can be
used to get increasingly sharper results, determining the value of lim supt↓0(C0
t+τs−s)/f(t) for
functions fcontaining powers of iterated logarithms, when stronger control over the densities of
the marginals of Xis available. Such refinements are possible when Xis a stable process cf.

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 10
Section 5. In particular, we may prove the following law of iterated logarithm given in [8, p. 54]
for a Cauchy process Xwith density x7→ pX(t, x) at time t > 0: for any s∈Rand the function
f(t) = (log log log(1/t))/log(1/t), we have limsupt↓0(C0
t+τs−s)/f(t)=1/pX(1, s) a.s. ♦
2.3. Regime (IS): upper functions at time 0.Throughout this subsection we assume Xhas
infinite variation, equivalent to lim inft↓0C0
t=−∞ a.s. [2, Sec. 1.1.2]. The following theorem describes
the upper fluctuations of C0
tas t↓0.
Theorem 2.9. Let fbe continuous and increasing with f(0) = 0 = limc↓0lim supt↓0f(ct)/f(t)and
f(t)≤1 = f(1) for t∈(0,1]. Let c > 0, denote F(t):=t/f(t)for t > 0and consider the conditions
Z1
0
P(Xt≤ −cF (t))dt
t<∞,(7)
Z1
0
E[(Xt/F (t))21{−2F(t)<Xt≤−t}]dt
t<∞,(8)
2nZ2−n
0
P(−t/f(2−n)≥Xt>−2F(t/2))dt→0,as n→ ∞.(9)
Then the following statements hold.
(i) If (7)–(9)hold for c= 1 and fis concave, then lim supt↓0|C0
t|f(t)=0a.s.
(ii) If (7)fails for all c > 0, then lim supt↓0|C0
t|f(t) = ∞a.s.
(iii) If lim supt↓0|C0
t|f(t)<1a.s., then (7)holds for any c > 1.
Some remarks are in order.
Remark 2.10.
(a) Any continuous regularly varying function fof index r > 0 satisfies the assumption in the theorem,
see Remark 2.3(a) above.
(b) The proof of Theorem 2.9 is based on the analysis of the upper fluctuations of the vertex time
τ−1/u as u↓0. The interpretation and purpose of conditions (7)–(9) are analogous to those of
conditions (2)–(4), respectively, see Remark 2.3(b) above.
(c) Note that (9) holds if R1
0P(−2F(2−nt/2) < X2−nt≤ −tF (2−n))dt→0 as n→ ∞, which, by the
dominated convergence theorem, is the case if P(−2F(u/2) < Xu≤ −tF (u/t)) →0 as u↓0 for
a.e. t∈(0,1).
(d) The assumed concavity of fin part (ii) can be dropped by modifying assumption (8) into a
condition involving the inverse of f(cf. Corollary 3.7 and Proposition 3.5). We do not make
this explicit in the statement of Theorem 2.9 because the functions of interest in this context are
typically concave. ♦
2.3.1. Simple sufficient conditions for the assumptions of Theorem 2.9.The tail probabilities of Xt
appearing in the assumptions of Theorem 2.9 are not analytically available in general. In this subsec-
tion we present sufficient conditions, in terms of the generating triplet (γ, σ2, ν) of X, implying the
assumptions in (7)–(9) of Theorem 2.9. Recall that σ2(ε) = σ2+R(−ε,ε)x2ν(dx) for ε > 0, and define:
(10) γ(ε):=γ−Z(−1,1)\(−ε,ε)
xν(dx), ν (ε):=ν(R\(−ε, ε)),for all ε > 0.
Let fand Fbe as in Theorem 2.9 and note that F(t)∈(0,1] since fis concave with f(1) = 1. The
inequalities in Lemma A.1 (with p= 2, ε=F(t)∈(0,1] and K=cF (t)), applied to P(|Xt| ≥ cF (t))
and E[min{X2
t,4F(t)2}]≥E[X2
t1{|Xt|≤2F(t)}], show that the condition
(11) Z1
0F(t)−2γ2(F(t))t+σ2(F(t))+ν(F(t))dt < ∞,

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 11
implies (7)–(8). Similarly, by Remark 2.10(c) and Lemma A.1, the following condition implies (9):
(12) F(t)−2γ2(F(t))t+σ2(F(t))+ν(F(t))t→0,as t↓0.
These simplifications lead to the following corollary.
Corollary 2.11. Suppose ν(ε) + ε−2σ2(ε) + ε−1|γ(ε)|=O(ε−β)as ε↓0for some β∈[1,2]
and, as before, let F(t) = t/f(t). If we have F(t)−βt→0as t→0and R1
0F(t)−βdt < ∞, then
lim supt↓0|C0
t|f(t)=0a.s.
Proof. By virtue of Theorem 2.9(i), it suffices to verify (11) and (12). By assumption, we have
[F(t)−2σ2(F(t)) + ν(F(t))]t=O(F(t)−βt) and F(t)−2γ(F(t))2t2=O((F(t)−βt)2), which tend to 0 as
t↓0, implying (12). Condition (11) follows similarly, completing the proof. 
Define the Blumenthal–Getoor index βBG ∈[0,2] of X[11] as follows:
(13) βBG := inf{q∈[0,2] : Iq<∞},where Iq:=Z(−1,1)\{0}|x|qν(dx), q > 0.
Note that, in our setting, Xhas infinite variation and hence βBG ≥1. Since Iβ<∞for any β > βBG,
[18, Lem. 1] shows that βsatisfies the assumptions of Corollary 2.11. Hence lim supt↓0|C0
t|tp= 0 a.s.
for any p > 1−1/βBG ∈[0,1/2] by Corollary 2.11.
Stronger results are possible when stronger conditions are imposed on the law of X. For instance,
for stable processes we have the following consequence of Theorem 2.9.
Corollary 2.12. Let Xbe an α-stable process with α∈[1,2). Then the following statements hold.
(a) If t7→ t−1/αF(t)is bounded as t↓0, then lim supt↓0|C0
t|f(t) = ∞a.s.
(b) If t−1/αF(t)→ ∞ as t↓0and Xis not spectrally positive, then the limit lim supt↓0|C0
t|f(t)is
equal to ∞(resp. 0) a.s. if the integral R1
0F(t)−αdtis infinite (resp. finite).
Proof. The scaling property of Xgives P(Xt≤ −cF (t)) = P(X1≤ −ct−1/αF(t)) for any c, t > 0. If
t7→ t−1/αF(t) is bounded, then lim inf t↓0P(Xt≤ −cF (t)) >0 making (7) fail for all c > 0. In that
case, we have lim supt↓0|C0
t|f(t) = ∞a.s. by Theorem 2.9(ii), proving part (a).
To prove part (b), suppose Xis not spectrally positive and let t−1/αF(t)→ ∞ as t↓0. Then
xαP(X1≤ −x) converges to a positive constant as x→ ∞, implying the following equivalence:
R1
0t−1P(Xt≤ −ct−1/αF(t))dt < ∞if and only if R1
0F(t)−αdt < ∞, where we note that the last
integral does not depend of c > 0. If R1
0F(t)−αdt < ∞, then (11)–(12) hold and Theorem 2.9(i) gives
lim supt↓0|C0
t|f(t) = 0 a.s. If instead R1
0F(t)−αdt=∞, then R1
0t−1P(Xt≤ −ct−1/αF(t))dt=∞for
all c > 0, so Theorem 2.9(ii) implies that lim supt↓0|C0
t|f(t) = ∞a.s., completing the proof. 
For Cauchy process (i.e. α= 1), Corollary 2.12 contains the dichotomy in [8, Cor. 3] for the upper
functions of C0at time 0. We note here that results analogous to Corollary 2.12 can be derived for
a spectrally positive stable process X(and for Brownian motion), using the exponential (instead of
polynomial) decay of the probability P(X1≤x) in xas x→ −∞, see [34, Thm 4.7.1].
2.4. Regime (IS): lower functions at time 0.As explained before, obtaining fine conditions for
the lower fluctuations of C0is more delicate than in the case of upper fluctuations of C0at 0. The
main reason is that the proof of Theorem 2.13 requires strong control on the Laplace exponent Φu(w)
of τu, defined in (1), as w→ ∞ and u→ −∞. This in turn requires sharp two-sided estimates on the
negative tail probability P(Xt≤ut) as a function of (u, t) as u→ −∞ and t↓0 jointly.
Due to the necessity of such strong control, in the following result we assume X∈ Zα,ρ for some
α > 1. In other words, we assume there exist some normalising function gthat is regularly varying

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 12
at 0 with index 1/α and an α-stable process (Zs)s∈[0,T ]with ρ=P(Z1>0) ∈(0,1) such that
(Xut/g(t))u∈[0,T ]
d
−→ (Zu)u∈[0,T]as t↓0. Recall that G(t) = t/g(t) for t > 0.
Theorem 2.13. Let X∈ Zα,ρ for some α∈(1,2] (and hence ρ∈(0,1)). Let f: (0,1) →(0,∞)be
given by f(t):=G(tlogp(1/t)), for some p∈Rand all t∈(0,1). Then the following statements hold:
(i) if p > 1/(1 −ρ), then lim inft↓0|C0
t|f(t) = ∞a.s.,
(ii) if p < 1/(1 −ρ), then lim inft↓0|C0
t|f(t) = 0 a.s.
Remark 2.14.
(a) The assumption X∈ Zα,ρ for some α > 1 implies that Xis of infinite variation. Note that
the function fin Theorem 2.13 is regularly varying at 0 with index 1 −1/α. The ‘negativity’
parameter 1 −ρ= limt↓0P(Xt<0) ∈(0,1) is a nontrivial function of the L´evy measure of
X. The fact that 1 −ρfeatures as a boundary point in the power of the logarithmic term in
Theorem 2.13 indicates that the lower fluctuations of C0at time 0 depends in a subtle way on
the characteristics of X. Such dependence is, for instance, not present for the upper fluctuations
of C0at time 0 when Xis α-stable, see Corollary 2.12 above. Indeed, for an α-stable process X,
Theorem 2.13 and Corollary 2.12(b) show that lim inft↓0|C0
t|f(t) = 0 and lim supt↓0|C0
t|f(t) = ∞
a.s. for f(t) = t1−1/α logq(1/t) and any q∈[−1/α, (1 −1/α)/(1 −ρ)), demonstrating the gap
between the lower and upper fluctuations of C0at time 0.
(b) The case where Xis attracted to Cauchy process with α= 1 is expected to hold for the functions f
in Theorem 2.13. As explained in Remark 2.8(a) above, many cases arise, with even some abrupt
processes being attracted to Cauchy process (see [2, Ex. 2.2]). We again stress that, in this case,
our methodology can be used to obtain a description of the upper fluctuations of C0at time 0 via
Theorem 3.3 and two-sided estimates, analogous to Lemma 4.6, of the Laplace exponent Φ in (1)
of the vertex time process. In the interest of brevity, we omit the details of such extensions.
(c) As with Theorem 2.7 above (see Remark 2.8(b)), the boundary case p= 1/(1−ρ) in Theorem 2.13
can be analysed along similar lines. In fact, our methods can be used to get increasingly sharper
results for the lower fluctuations of C0at time 0 when stronger control over the negative tail
probabilities for the marginals Xis available. Such improvements are possible, for instance, when
Xis α-stable. We decided to leave such results for future work in the interest of brevity. For
completeness, however, we mention that the following law of iterated logarithm proved in [8, Cor. 3]
can also be proved using our methods (see Example 4.3 below): lim inf t↓0|C0
t|f(t) = pX(1,0) a.s.,
where x7→ pX(t, x) is the density of Xt.♦
2.5. Upper and lower function of the L´evy path at vertex times. In this section we establish
consequences for the lower (resp. upper) fluctuations of the L´evy path at vertex time τs(resp. time
0) in terms of those of C0. Recall Xt−:= limu↑tXufor t > 0 (and X0−:=X0) and define ms:=
min{Xτs, Xτs−}for s∈ L+(S).
Lemma 2.15. Suppose s∈ L+(S). Let the function f: [0,∞)→[0,∞)be continuous and increasing
and define the function ˜
f(t):=Rt
0f(u)du,t≥0. Then the following statements hold for any M > 0.
(i) If lim inft↓0(C0
t+τs−s)/f(t)> M a .s. then lim inf t↓0(Xt+τs−ms−st)/˜
f(t)≥Ma.s.
(ii) If lim supt↓0(C0
t+τs−s)/f(t)< M a.s. then lim inf t↓0(Xt+τs−ms−st)/˜
f(t)≤Ma.s.
The proof of Lemma 2.15 is pathwise. The lemma yields the following implications
(i) lim inft↓0(C0
t+τs−s)/f(t) = ∞=⇒lim inft↓0(Xt+τs−ms−st)/˜
f(t) = ∞,
(ii) lim supt↓0(C0
t+τs−s)/f(t) = 0 =⇒lim inft↓0(Xt+τs−ms−st)/˜
f(t) = 0.

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 13
The upper fluctuations of Xat vertex time τscannot be controlled via the fluctuations of C0since the
process may have large excursions away from its convex minorant between contact points. Moreover,
the limits lim inf t↓0(C0
t+τs−s)/f(t) = 0 or limsupt↓0(C0
t+τs−s)/f(t) = ∞, do not provide sufficient
information to ascertain the value of the lower limit lim inf t↓0(Xt+τs−ms−st)/˜
f(t), since this limit
may not be attained along the contact points between the path and its convex minorant.
Theorems 2.2 a nd 2.7 give sufficient conditions, in terms of the law of X, for the assumptions in
Lemma 2.15 to hold. This leads to the following corollaries.
Corollary 2.16. Let s∈ L+(S)and let fbe a continuous and increasing function with f(0) = 0 =
limc↓0lim supt↓0f(ct)/f(t),f(1) = 1 and f(t)≤1for t∈(0,1]. If conditions (2)–(4)hold for c= 1,
then lim inft↓0(Xt+τs−ms−st)/˜
f(t) = ∞a.s. where we denote e
f(t):=Rt
0f(u)du.
Denote by $(t):=t−1/αg(t) the slowly varying (at 0) component of the normalising function gof
a process in the class Zα,ρ. Recall that G(t) = t/g(t) for t > 0.
Corollary 2.17. Let X∈ Zα,ρ for some α∈(0,1) and ρ∈(0,1]. Given p∈R, denote ˜
f(t):=
Rt
0G(ulogp(u−1))−1dufor t > 0. Then the following statements hold for s=γ0.
(i) If p > 1/ρ, then lim inft↓0(Xt+τs−ms−st)/˜
f(t) = 0 a.s.
(ii) If α∈(1/2,1),p < −α/(1 −α)and ($(c/t)/$(1/t)−1) log log(1/t)→0as t↓0for some
c∈(0,1), then lim inft↓0(Xt+τs−ms−st)/˜
f(t) = ∞a.s.
(iii) If α∈(0,1/2], then lim inft↓0(Xt+τs−ms−st)/tq=∞a.s. for any q > 1/α ≥2.
Remark 2.18.
(a) The function ˜
fis regularly varying at 0 with index 1/α. This makes conditions in Corollary 2.17
nearly optimal in the following sense: the polynomial rate in all three cases is either 1/α (cases
(i) and (ii) in Corollary 2.17) or arbitrarily close to it (case (iii) in Corollary 2.17). If α > 1/2,
then the gap is in the power of the logarithm in the definition of ˜
f.
(b) When the natural drift γ0= 0, Corollary 2.17 describes the lower fluctuations (at time 0) of
the post-minimum process X→= (X→
t)t∈[0,T −τ0]given by X→
t:=Xt+τ0−m0(note that m0=
inft∈[0,T ]Xt). The closest result in this vein is [36, Prop. 3.6] where Vigon shows that, for any
infinite variation L´evy process Xand r > 0, we have lim inft↓0X→
t/t ≥ra.s. if and only if
R1
0P(Xt/t ∈[0, r])t−1dt < ∞. Our result considers non-linear functions and a large class of finite
variation processes.
(c) By [35, Thm 2], the assumption X∈ Zα,ρ and γ0= 0 implies that the post-minimum process,
conditionally given τ0, is a L´evy meander. Hence, Corollary 2.17 also describes the lower functions
of the meanders of L´evy processes in Zα,ρ. A similar remark applies to the results in Corollary 2.16.
♦
When Xhas infinite variation, the process Xand Ctouch each other infinitely often on any
neighborhood of 0 (see [2]), leading to the following connection in small time between the paths of X
and its convex minorant C.
Lemma 2.19. Let the function f: [0,∞)→[0,∞)be continuous and increasing with f(0) = 0 and
finite ˜
f(t):=Rt
0f(u)−1du,t≥0. Then the following statements hold for any M > 0.
(i) If lim supt↓0|C0
t|f(t)< M a.s., then lim supt↓0(−Xt)/˜
f(t)≤Ma.s.
(ii) If lim inft↓0|C0
t|f(t)> M a.s., then lim supt↓0(−Xt)/˜
f(t)≥Ma.s.
Theorem 2.9 and the corollaries thereafter give sufficient explicit conditions for the assumption
in Lemma 2.19(i) to hold. Similarly, Theorem 2.13 gives a fine class of functions fsatisfying the

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 14
assumption in Lemma 2.19(ii) for a large class of processes. Such conclusions on the fluctuations of
the L´evy path of Xwould not be new as the fluctuations of Xat 0 are already known, see [12,32,
33]. In particular, the upper functions of Xand −Xat time 0 were completely characterised in [32]
in terms of the generating triplet of X. Let us comment on some two-way implications of our results,
the literature and Lemma 2.19.
Remark 2.20.
(a) By [22], the assumption in Theorem 2.9(ii) implies that lim supt↓0|Xt|/F (t) = ∞a.s. where we
recall that F(t) = t/f(t). Similarly, by [22], if lim supt↓0|Xt|/F (t) = ∞a.s. then the assumption
in Theorem 2.9(ii) must hold for either Xor −X, which, by time reversal, implies that at least one
of the limits lim supt↓0|C0
t|f(t) or lim supt↓0|C0
T−t|f(t) is infinite a.s. This conclusion is similar
to that of Lemma 2.19, the main difference being the use of either ˜
for F. Note however, that if
fis regularly varying with index different from 1, then [9, Thm 1.5.11] implies limt↓0˜
f(t)/F (t)∈
(0,∞).
(b) The contrapositive statements of Lemma 2.19 give information on C0in terms of −X. Indeed, if we
have lim supt↓0(−Xt)/˜
f(t)>0, then lim supt↓0|C0
t|f(t)>0. Similarly, if lim supt↓0(−Xt)/˜
f(t)<
∞, then lim inft↓0|C0
t|f(t)<∞.♦
The connections between the fluctuations of Xand those of C0at time 0 are intricate. Although
the one-sided fluctuations of Xat 0 were essentially characterised in [32, Thm 3.1], its combination
with Lemma 2.19 is not sufficiently strong to obtain conditions for any of the following statements:
lim supt↓0|C0
t|f(t) = 0, lim supt↓0|C0
t|f(t)>0, lim inf t↓0|C0
t|f(t)<∞or lim inf t↓0|C0
t|f(t) = ∞a.s.
3. Small-time fluctuations of non-decreasing additive processes
Consider a pure-jump right-continuous non-decreasing additive (i.e. with independent and possibly
non-stationary increments) process Y= (Yt)t≥0with Y0= 0 a.s. and its mean jump measure Π(dt, dx)
for (t, x)∈[0,∞)×(0,∞), see [20, Thm 15.4]. Then, by Campbell’s formula [20, Lem. 12.2], its Laplace
transform satisfies
(14) Ee−uYt=e−Ψt(u),where Ψt(u):=Z(0,∞)
(1 −e−ux)Π((0, t],dx),for any u≥0.
Let Lt:= inf{u > 0 : Yu> t}for t≥0 (with convention inf ∅=∞) denote the right-continuous
inverse of Y. Our main objective in this section is to describe the upper and lower fluctuations of L,
extending known results for the case where Yhas stationary increments (making Ya subordinator)
in which case Π(dt, dx) = Π((0,1],dx)dtfor all (t, x)∈[0,∞)×(0,∞) (see e.g. [7, Thm 4.1]).
3.1. Upper functions of L.The following theorem is the main result of this subsection.
Theorem 3.1. Let f: (0,1) →(0,∞)be increasing with limt↓0f(t) = 0 and φ: (0,∞)→(0,∞)
be decreasing with limu→∞ φ(u) = 0. Let the positive sequence (θn)n∈Nsatisfy limn→∞ θn=∞and
define the associated sequence (tn)n∈Ngiven by tn:=φ(θn)for any n∈N.
(a) If P∞
n=1 exp(θntn−Ψf(tn)(θn)) <∞then lim supt↓0Lt/f(t)≤lim supn→∞ f(tn)/f(tn+1)a.s.
(b) If limu→∞ φ(u)u=∞,P∞
n=1[exp(−Ψf(tn)(θn)) −exp(−θntn)] = ∞and P∞
n=1 Ψf(tn+1)(θn)<∞,
then lim supt↓0Lt/f(t)≥1a.s.
Remark 3.2.
(a) Theorem 3.1 plays a key role in the proofs of Theorems 2.7 and 2.13. Before applying Theorem 3.1,
one needs to find appropriate choices of the free infinite-dimensional parameters hand (θn)n∈N.

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 15
This makes the application of Theorem 3.1 hard in general and is why, in Theorems 2.7 and 2.13,
we are required to assume that Xlies in the domain of attraction of an α-stable process.
(b) If Yhas stationary increments (making Ya subordinator), the proof of [7, Thm 4.1] follows from
Theorem 3.1 by finding an appropriate function fand sequences (θn)n∈N(done in [7, Lem. 4.2 &
4.3]) satisfying the assumptions of Theorem 3.1. In this case, the function fis given in terms of
the single-parameter Laplace exponent Ψ1, see details in [7, Thm 4.1]. ♦
Proof of Theorem 3.1.(a) Since Lis the right-inverse of Y, we have {Ltn> f(tn)}={tn≥Yf(tn)}
for n∈N. Using Chernoff ’s bound (Markov’s inequality), we obtain
Ptn≥Yf(tn)≤eθntnEexp −θnYf(tn)= exp(θntn−Ψf(tn)(θn)),for all n≥1.
The assumption P∞
n=1 exp(θntn−Ψf(tn)(θn)) <∞thus implies P∞
n=1 P(Ltn> f(tn)) <∞. Hence,
the Borel–Cantelli lemma yields lim supn→∞ Ltn/f (tn)≤1 a.s. Since Lis non-decreasing and (tn)n∈N
is decreasing monotonically to zero, we have
lim sup
t↓0
Lt
f(t)≤lim sup
n→∞
sup
t∈[tn+1,tn]
Ltn
f(t)≤lim sup
n→∞
Ltn
f(tn)·lim sup
n→∞
f(tn)
f(tn+1)≤lim sup
n→∞
f(tn)
f(tn+1)a.s.,
which gives (a).
(b) It suffices to establish that the following limits hold: liminfn→∞(Yf(tn)−Yf(tn+1 ))/tn≤1 a.s.
and lim supn→∞ Yf(tn+1)/tn≤δa.s. for any δ > 0. Indeed, by taking δ↓0 along a countable sequence,
the second limit gives lim supn→∞ Yf(tn+1 )/tn= 0 a.s. and hence lim inf n→∞ Yf(tn)/tn≤1 a.s. For
any t > 0 with Yf(t)≤twe have Lt> f(t). Since the former holds for arbitrarily small values of t > 0
a.s., we obtain lim supt↓0Lt/f (t)≥1 a.s.
We will prove that liminfn→∞(Yf(tn)−Yf(tn+1 ))/tn≤1 a.s. and lim supn→∞ Yf(tn+1 )/tn≤δa.s.
for any δ > 0, using the Borel–Cantelli lemmas. Applying Markov’s inequality, we obtain the upper
bound P(Yt> s)≤(1 −e−θs )−1E[1 −e−θYt] for all t, s, θ > 0, implying
PYf(tn)≤tn≥exp(−Ψf(tn)(θn)) −exp(−θntn)
1−exp(−θntn),for all n≥1.
Since θntn=θnφ(θn)→ ∞ as n→ ∞, the denominator of the lower bound in the display above
tends to 1 as n→ ∞, and hence the assumption P∞
n=1[exp(−Ψf(tn)(θn)) −exp(−θntn)] = ∞implies
P∞
n=1 P(Yf(tn)< tn) = ∞. Since Yhas non-negative independent increments and
∞
X
n=1
P(Yf(tn)−Yf(tn+1)< tn)≥
∞
X
n=1
P(Yf(tn)< tn) = ∞,
the second Borel–Cantelli lemma yields lim infn→∞(Yf(tn)−Yf(tn+1 ))/tn≤1 a.s.
To prove the second limit, use Markov’s inequality and the elementary bound 1 −e−x≤xto get
PYf(tn+1)> δtn≤E[1 −exp(−θnYf(tn+1))]
1−exp(−δθntn)=1−exp(−Ψf(tn+1 )(θn))
1−exp(−δθntn)≤Ψf(tn+1 )(θn)
1−exp(−δθntn),
for all n∈N. Again, the denominator tends to 1 as n→ ∞ and the assumption P∞
n=1 Ψf(tn+1)(θn)<∞
implies P∞
n=1 P(Yf(tn+1)> δtn)<∞. The Borel–Cantelli lemma implies lim supn→∞ Yf(tn+1 )/tn≤δ
a.s. and completes the proof. 
3.2. Lower functions of L.To describe the lower fluctuations of L, it suffices to describe the upper
fluctuations of Y. The following result extends known results for subordinators (see, e.g. [15, Thm

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 16
(15)
(16)
(19)
(18)
(17)
(20)
(21) (22)
(23)
h−1concave
h−1concave
h−1concave
h−1concave
Figure 2. A graphical representation of the implications in Theorem 3.3 and Propo-
sition 3.5.
1]). Given a continuous increasing function hwith h(0) = 0 and h(1) = 1, consider the following
statements, used in the following result to describe the upper fluctuations of Y:
lim sup
t↓0
Yt/h(t) = 0,a.s.,(15)
lim sup
t↓0
Yt/h(t)<1,a.s.,(16)
Π({(t, x) : t∈(0,1], x ≥h(t)})<∞,(17)
Z(0,1]×(0,1)
x2
h(t)21{2h(t)>x}Π(dt, dx)<∞,(18)
2nZ(0,h−1(2−n)]×(0,2−n)
x1{2h(t)>x}Π(dt, dx)↓0,as n→ ∞,and(19)
Z(0,1]×(0,1)
x
h(t)1{2h(t)>x}Π(dt, dx)<∞.(20)
Theorem 3.3. Let hbe continuous and increasing with h(0) = 0 and h(1) = 1. Then the following
implications hold:
(a) (15) =⇒(16) =⇒(17),(b) (17)–(19) =⇒(15),(c) (20) =⇒(18)–(19).
Remark 3.4.If his as in Theorem 3.3 and Π({(t, x) : t∈(0,1], x ≥ch(t)}) = ∞for all c > 0, then it
follows from the negation of Theorem 3.3(a) that lim sup↓0Yt/h(t) = ∞a.s. ♦
In the description of the lower fluctuations of L, we are typically given the function h−1directly
instead of h. In those cases, the conditions in Theorem 3.3 may be hard to verify directly (see e.g. the
proof of Theorem 2.9(i)). To alleviate this issue, we introduce alternative conditions describing the
upper fluctuations of Yin terms of the function h−1. However, this requires the additional assumption
that h−1is concave, see Proposition 3.5 below. Consider the following conditions on h−1:
Z(0,1]×(0,1)
h−1(x)2
t21{2t≥h−1(x)}Π(dt, dx)<∞,(21)
2nZ(0,2−n]×(0,h(2−n))
h−1(x)1{2t≥h−1(x)}Π(dt, dx)↓0,as n→ ∞,and(22)
Z(0,1]×(0,1)
h−1(x)
t1{2t≥h−1(x)}Π(dt, dx)<∞.(23)

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 17
Proposition 3.5. Let hbe convex and increasing with h(0) = 0 and h(1) = 1. Then the following
implications hold:
(a) (21) =⇒(18),(b) (23) =⇒(21)–(22),(c) (17)and (21)–(22) =⇒(15).
The relation between the assumptions of Theorem 3.3 and Proposition 3.5 (concerning hand h−1)
is described in Figure 2. The following elementary result explains how the upper fluctuations of Y
(described by Theorem 3.3) are related to the lower fluctuations of L.
Lemma 3.6. Let hbe a continuous increasing function with h(0) = 0 and denote by h−1its inverse.
Then the following implications hold for any c > 0:
(a) lim inft↓0Lt/h−1(t/c)>1 =⇒lim supt↓0Yt/h(t)≤c,
(b) lim supt↓0Yt/h(t)< c =⇒lim inft↓0Lt/h−1(t/c)≥1.
Proof. The result follows from the implications Lu> t =⇒u≥Yt=⇒Lu≥tfor any t, u > 0.
Indeed, if lim inf u↓0Lu/h−1(u/c)>1 then Lu> h−1(u/c) for all sufficiently small u > 0 implying that
Yt≤ch(t) for all sufficiently small t > 0 and hence lim supt↓0Yt/h(t)≤c. This establishes part (a).
Part (b) follows along similar lines. 
A combination of Lemma 3.6, Theorem 3.3, Proposition 3.5 and Remark 3.4 yield the following
corollary.
Corollary 3.7. Let hbe a continuous and increasing function with h(0) = 0 and h(1) = 1 such that
limc↓0lim supt↓0h−1(ct)/h−1(t)=0. Then the following results hold:
(i) If lim inft↓0Lt/h−1(t/c)>1a.s. for some c∈(0,1) then (17)holds.
(ii) Suppose (17)–(19)hold, then lim inft↓0Lt/h−1(t) = ∞a.s.
(ii’) Suppose his convex and conditions (17)and (21)–(22)hold, then lim inf t↓0Lt/h−1(t) = ∞a.s.
(iii) If Π({(t, x) : t∈(0,1], x ≥ch(t)}) = ∞for all c > 0then lim inft↓0Lt/h−1(t)=0a.s.
To prove Theorem 3.3 we require the following lemma. For all t≥0 denote by ∆t:=Yt−Yt−the
jump of Yat time t, so that Yt=Pu≤t∆usince Yis a pure-jump additive process. We also let N
denote the Poisson jump measure of Y, given by N(A):=|{t: (t, ∆t)∈A}| for A⊂[0,∞)×(0,∞)
and note that its mean measure is Π(dt, dx).
Lemma 3.8. Let hbe continuous and increasing with h(0) = 0 and h(1) = 1. Assume (17)–(19)hold,
then lim supt↓0Yt/h(t) = lim supt↓0Yh−1(t)/t = 0 a.s.
Proof. For all n∈N, we let Bn:= [2−n,∞) and set Cn:=h−1((2−n−1,2−n]) ×Bn. Then we have
X
n∈N
P(N(Cn)≥1) = X
n∈N1−e−Π(Cn)≤X
n∈N
Π(Cn),
by the definition of Nand the inequality 1 −e−x≤x. Note that Pn∈NΠ(Cn)<∞by (17), since
X
n∈N
Π(Cn)≤Π({(t, x) : t∈[0,1], x ≥h(t)})<∞.
By the Borel–Cantelli lemma, there exists some n0∈Nwith N(h−1((2−n−1,2−n]) ×Bn) = 0 a.s.
for all n≥n0. By the mapping theorem, the random measure Nh(A×B):=N(h−1(A)×B) for
any measurable A, B ⊂[0,∞), is a Poisson random measure with mean measure Πh(A×B):=
Π(h−1(A), B). Note that Yh−1(t)=R(0,h−1(t)]×(0,∞)xN (du, dx) = R(0,t]×(0,∞)xNh(du, dx) for t≥0
and, for any n≥n0and t∈(2−n−1,2−n], we have |Yh−1(t)/t| ≤ ζn:= 2n+1 P∞
m=nξm, where
ξm:=Z(2−m−1,2−m]×(0,2−m)
xNh(du, dx), m ∈N.

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 18
To complete the proof, it suffices to show that ζn↓0 a.s. as n→ ∞. Fubini’s theorem yields
2−n−1E[ζn] =
∞
X
m=nZ(2−m−1,2−m]×(0,2−m)
xΠh(du, dx)
=Z(0,2−n]×(0,2−n)
x
∞
X
m=n
1{x<2−m}1{u≤2−m<2u}Πh(du, dx)
≤Z(0,2−n]×(0,2−n)
x1{2u>x}Πh(du, dx) = Z(0,h−1(2−n)]×(0,2−n)
x1{2h(v)>x}Π(dv, dx).
By assumption (19), we deduce that E[ζn]↓0 as n→ ∞. Similarly, note that
Var(ζn)=4n+1
∞
X
m=nZ(2−m−1,2−m]×(0,2−m)
x2Πh(du, dx),
and hence, by Fubini’s theorem and assumption (18), we have
∞
X
n=1
Var(ζn) =
∞
X
m=1
m
X
n=1
4n+1 Z(2−m−1,2−m]×(0,2−m)
x2Πh(dt, dx)
≤
∞
X
m=1
4m+2 Z(2−m−1,2−m]×(0,2−m)
x2Πh(du, dx)
=Z(0,1]×(0,1)
x2
∞
X
m=1
4m+21{x<2−m}1{u<2−m<2u}Πh(du, dx)
≤42Z(0,1]×(0,1)
x2
u21{2u>x}Πh(du, dx)
= 42Z(0,h−1(1)]×(0,1)
x2
h(v)21{2h(v)>x}Π(dv, dx)<∞.
Thus, we find that the sum P∞
n=1(ζn−E[ζn])2has finite mean equal to P∞
n=1 Var(ζn)<∞and is
thus finite a.s. Hence, the summands must tend to 0 a.s. and, since E[ζn]→0, we deduce that ζn↓0
a.s. as n→ ∞.
Proof of Theorem 3.3.It is obvious that (15) implies (16). If (16) holds, then Yt< h(t) for all
sufficiently small t. Thus, the path bound Yt≥∆timplies P(N({(t, x) : t∈[0,1], x > h(t)})<∞) =
1 and hence (17). By Lemma 3.8, conditions (17)–(19) imply (15), so it remains to show that (20)
implies (18) and (19).
It is easy to see that (20) implies (18). Moreover, if (20) holds, then
2nZ(0,h−1(2−n)]×(0,2−n)
x1{2h(t)>x}Π(dt, dx)
≤Z(0,h−1(1)]×(0,1)
x
h(t)1{2h(t)>x}1(0,h−1(2−n)]×(0,2−n)(t, x)Π(dt, dx),
where the upper bound is finite for all n∈Nand tends to 0 as n→ ∞ by the monotone convergence
theorem, implying (19). 
Proof of Proposition 3.5.Since h−1is concave with h−1(0) = 0, then x7→ h−1(x)/x is decreasing, so
the condition h(t)> x/2 implies (x/2)/h(t)≤h−1(x/2)/t ≤h−1(x)/t. The inequality h−1(x)/x ≤
h−1(x/2)/(x/2) implies that {(t, x):2h(t)> x} ⊂ {(t, x):2t > h−1(x)}, proving the first claim:
(21) implies (18).

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 19
Since h−1is concave with h−1(0) = 0, it is subadditive, implying
ζt:=X
u≤t
h−1(∆u)≥h−1(Yt).
Since lim supt↓0ζt/t ≤cimplies lim supt↓0Yt/h(ct)≤1 for c > 0 and his a convex function, it suffices
to show that lim supt↓0ζt/t = 0 a.s. Note that ζis an additive process with jump measure Π(dt, h(dx)).
Applying Theorem 3.3 to ζwith the identity function yields the result, completing the proof. 
Remark 3.9.We now show that, when the increments of Yare stationary (making Ya subordinator),
Theorem 3.3 gives a complete characterisation of the upper functions of Y, recovering [15, Thm 1] (see
also [7, Prop. 4.4]). This is done in two steps.
Suppose his convex and Yhas stationary increments with mean jump measure Π(dt, dx) =
Π((0,1],dx)dt. Then h−1is concave and the additive process e
Yt:=Ps≤th−1(∆s)≥h−1(Yt) has
mean jump measure Π(dt, h(dx)), making it a subordinator. Theorem 3.3 applied to e
Yand the iden-
tity function makes all conditions (17)–(19) equivalent to R(0,1) h−1(x)Π((0,1],dx)<∞and therefore,
by Theorem 3.3, also equivalent to the condition lim supt↓0e
Yt/t = 0 a.s.
Note that condition (17) for e
Yand the identity function coincides with condition (17) for Yand h.
This equivalence, together with the fact that the limit lim supt↓0e
Yt/t = 0 implies lim supt↓0Yt/h(t) = 0,
shows that both limits are either 0 a.s. or positive a.s. jointly. Thus, lim supt↓0Yt/h(t) = 0 a.s. if and
only if R(0,1) h−1(x)Π((0,1],dx)<∞and, if the latter condition fails, then lim supt↓0Yt/h(t) = ∞a.s.
by Remark 3.4. This is precisely the criterion given in [15, Thm 1] (see also [7, Prop. 4.4]). ♦
Remark 3.9 shows that condition (17) perfectly describes the upper fluctuations of Ywhen Y
has stationary increments, making conditions (18) & (19) appear superfluous. These conditions are,
however, not superfluous since (17) by itself cannot fully characterise the upper fluctuations of Y, as
the following example shows.
Example 3.10. Let Π(dt, dx) = Pn∈Nn−12nδ(2−n,2−n/n)(dt, dx), where δxdenotes the Dirac measure
at x, and consider the corresponding additive process Y(whose existence is ensured by [20, Thm 15.4]).
Since P(ξ≥µ)≥1/5 for every Poisson random variable ξwith mean µ≥2 [26, Eq. (6)], we
get Pn∈NP(N({(2−n,2−n/n)})≥2n/n) = ∞. The second Borel–Cantelli lemma then shows that
∆2−n≥1/n2i.o. Thus, Y2−n/2−n≥2n∆2−n≥2n/n2i.o., implying lim supt↓0Yt/t =∞a.s. even
when condition (17) holds. In fact, Π({(t, x) : t∈(0,1], x ≥ct})<∞for all c > 0. 4
4. The vertex time process and the proofs of the results in Section 2
We first recall basic facts about the vertex time process τ= (τs)s∈R. Fix a deterministic time
horizon T > 0, let Cbe the convex minorant of Xon [0, T ] with right-derivative C0and recall the
definition τs= inf{t > 0 : C0
t> s}for any slope s∈R. By the convexity of C, the right-derivative
C0is non-decreasing and right-continuous, making τa non-decreasing right-continuous process with
lims→−∞ τs= 0 and lims→∞ τs=T. Intuitively put, the process τfinds the times in [0, T ] at which
the slopes increase as we advance through the graph of the convex minorant t7→ Ctchronologically.
We remark that the vertex time process can be constructed directly from Xwithout any reference to
the convex minorant C, as follows (cf. [25, Thm 11.1.2]): for each slope s∈Rand time epoch t≥0,
define X(s)
t:=Xt−st,X(s)
t:= infu∈[0,t]X(s)
uand note τs= sup t∈[0, T ] : X(s)
t−∧X(s)
t=X(s)
T,
where X(s)
u−:= limv↑uX(s)
u−for u > 0 and X(s)
0−:=X(s)
0= 0. Put differently, subtracting a constant
drift sfrom the L´evy process X“rotates” the convex hull so that the vertex time τsbecomes the time
the minimum of X(s)during the time interval [0, T ] is attained.

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 20
4.1. The vertex time process over exponential times. Fix any λ > 0 and let Ebe an independent
exponential random variable with unit mean. Let b
C:= ( b
Ct)t∈[0,E/λ]be the convex minorant of X
over the exponential time-horizon [0, E/λ] and denote by bτthe right-continuous inverse of b
C0, i.e.
bτs:= inf{u∈[0, E/λ] : b
C0
u> s}for s∈R. Hence, in the remainder of the paper, the processes with
(resp. without) a ‘hat’ will refer to the processes whose definition is based on the path of Xon [0, E/λ]
(resp. [0, T ]), where Eis an exponential random variable with unit mean independent of Xand T > 0
is fixed and deterministic.
It is more convenient to consider the vertex time processes over an independent exponential time
horizon rather than the fixed time horizon T, as this does not affect the small-time behaviour of the
process (see Corollary 4.2 below), while making its law more tractable. Moreover, as we will see, to
analyse the fluctuations of b
C0over short intervals, it suffices to study those of bτ. By [17, Cor. 3.2], the
process bτhas independent but non-stationary increments and its Laplace exponent is given by
(24) E[e−ubτs] = e−Φs(u),where Φs(u):=Z∞
0
(1 −e−ut)e−λt P(Xt≤st)dt
t,
for all u≥0 and s∈R. The following lemma states that, after a vertex time, the convex minorants
Cand b
Cmust agree for a positive amount of time, see Figure 3for a pictorial description.
Lemma 4.1. For any s∈ L+(S), on the event {τs< E/λ ≤T}, we have τs=bτsand the convex
minorants Cand b
Cagree on and interval [0, τs+m]for a random m > 0. If Xis of infinite variation,
the functions Cand b
Cagree on an interval [0, m]for a random variable msatsifying 0< m ≤
min{T, E/λ}a.s.
Since the L´evy process Xand the exponential time Eare independent, P(τs< E/λ ≤T)>0.
Proof. The proof follows directly from the definition of the convex minorant of fas the greatest convex
function dominated by the path of fover the corresponding interval. Let fbe a measurable function
on [0, t] with piecewise linear convex minorant M(t). Then, for any vertex time v∈(0, t) of M(t)
and any u∈(v, t], the convex minorant M(u)of fon [0, u] equals M(t)over the interval [0, v]. The
result then follows since the condition s∈ L+(S) (resp. Xhas infinite variation) implies that there
are infinitely many vertex times immediately after τs(resp. 0). 
The following result shows that local properties of Cagree with those of b
C. Multiple extensions are
possible, but we opt for the following version as it is simple and sufficient for our purpose.
Corollary 4.2. Fix any measurable function f: (0,∞)→(0,∞).
(a) If s∈ L+(S), then the following limits are a.s. constants on [0,∞]:
lim sup
t↓0
C0
t+τs−s
f(t)= lim sup
t↓0b
C0
t+bτs−s
f(t)and lim inf
t↓0
C0
t+τs−s
f(t)= lim inf
t↓0b
C0
t+bτs−s
f(t).
(b) If Xis of infinite variation, then the following limits are a.s. constants on [0,∞]:
lim sup
t↓0
C0
t/f(t) = lim sup
t↓0b
C0
t/f(t)and lim inf
t↓0C0
t/f(t) = lim inf
t↓0b
C0
t/f(t).
Proof. We will prove part (a) for liminf, with the remaining proofs being analogous. First note that
the assumption s∈ L+(S) implies that (τu+s−τs)u≥0and the additive processes (bτu+s−bτs)u≥0have
infinite activity as u↓0. Thus, applying Blumenthal’s 0–1 law [20, Cor. 19.18] to (bτu+s−bτs)u≥0
(and using the fact that b
C0
bτs=sa.s.), implies that lim inft↓0(b
C0
t+bτs−s)/f(t) is a.s. equal to some
constant µin [0,∞]. Moreover, by the independence of the increments of bτs, this limit holds even

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 21
Figure 3. The picture shows a path of X(black) and its convex minorants C(red)
on [0, T ] and b
C(blue) on [0, E/λ]. Both convex minorants agree until time m, after
which they may behave very differently.
when conditioning on the value of bτs. Recall further that bτs=τson the event {τs< E/λ ≤T}by
Lemma 4.1. By Lemma 4.1 and the independence of Eand X, we a.s. have
0<P(τs< E/λ ≤T|τs) = Plim inf
t↓0(b
C0
t+bτs−s)/f(t) = µ, τs< E/λ ≤Tτs
=Plim inf
t↓0(C0
t+τs−s)/f(t) = µ, τs< E/λ ≤Tτs
=Plim inf
t↓0(C0
t+τs−s)/f(t) = µτsP(τs< E/λ ≤T|τs),
implying that lim inf t↓0(C0
t+τs−s)/f(t) = µa.s. 
By virtue of Corollary 4.2 it suffices to prove all the results in Section 2for b
Cinstead of C. This
allows us to use the independent increment structure of the right inverse bτof the right-derivative b
C0.
Example 4.3 (Cauchy process).If Xis a Cauchy process, then the Laplace exponent of bτufactorises
Φu(w) = P(X1≤u)R∞
0(1 −e−wt)e−λt t−1dtfor any u∈Rand w≥0. This implies that bτhas the
same law as a gamma subordinator time-changed by the distribution function u7→ P(X1≤u) =
1
2+1
πarctan(cu +µ) for some c > 0 and µ= tan(π(1
2−ρ)). This result can be used as an alternative
to [8, Thm 2], in conjunction with classical results on the fluctuations of a gamma process (see, e.g. [7,
Ch. 4]), to establish [8, Cor. 3] and all the other results in [8]. 4
The proofs of the results in Section 2are based on the results of Section 3: we will construct a non-
decreasing additive process Y= (Yt)t≥0, started at 0, in terms of bτand apply the results in Section 3
to Yand its inverse L= (Lu)u≥0. These proofs are given in the following subsections.
4.2. Upper and lower functions at time τs- proofs. Let s∈ L+(S). Fix any λ > 0 and let
Yu:=bτu+s−bτs,u≥0. Then the right-inverse Lu:= inf{t > 0 : Yt> u}of Yequals Lu=b
C0
u+bτs−s
for u≥0. Note that Yhas independent increments and (24) implies
(25) Ψu(w):=−log E[e−wYu] = Z∞
0
(1 −e−wt)Π((0, u],dt),for all w , u ≥0,
where Π(du, dt) = e−λtP((Xt−st)/t ∈du)t−1dtis the mean jump measure of Y.

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 22
Proof of Theorem 2.2.Since s∈ L+(S), all three parts of the result follow from a direct application
of Proposition 3.5 and Corollary 3.7 to the definitions of Yand Labove. 
To prove Theorem 2.7, we require the following two lemmas. The first lemma establishes some
general regularity for the densities of Xtas a function of tand the second lemma provides a strong
asymptotic control on the function Ψs(u) as s↓0 and u→ ∞. Recall that, when Xis of finite
variation, γ0= limt↓0Xt/t denotes the natural drift of X.
Lemma 4.4. Let X∈ Zα,ρ for some α∈(0,1) and ρ∈(0,1] and denote by gits normalising function.
(a) Define Qt:= (Xt−γ0t)/g(t), then Qthas an infinitely differentiable density ptsuch that ptand
each of its derivatives p(k)
tare uniformly bounded: supt∈(0,1] supx∈R|p(k)
t(x)|<∞for any k∈N∪{0}.
(b) Define e
Qt:=Xt/√t, then e
Qthas an infinitely differentiable density ˜ptsuch that ˜ptand each of its
derivatives ˜p(k)
tare uniformly bounded: supt∈[1,∞)supx∈R|˜p(k)
t(x)|<∞for any k∈N∪ {0}.
For two functions f1, f2: (0,∞)→(0,∞) we say f1(t)∼f2(t) as t↓0 if limt↓0f1(t)/f2(t) = 1.
Proof of Lemma 4.4.Part (a). We assume without loss of generality that g(t)≤1 for t∈(0,1], and
note that Qtis infinitely divisible. Denote by νQtthe L´evy measure of Qt, and note for A⊂Rthat
νQt(A) = tν(g(t)A) and
σ2
Qt(u):=Z(−u,u)
x2νQt(dx) = t
g(t)2Z(−ug(t),ug(t))
x2ν(dx) = t
g(t)2σ2(ug(t)),
for t∈(0,1] and u∈R\ {0}. The regular variation of ν(see [19, Thm 2]), Fubini’s theorem and
Karamata’s theorem [9, Thm 1.5.11(ii)] imply that, as u↓0,
σ2(u) = −Zu
0
x2ν(dx) = −Zu
0
2Zx
0
zdzν(dx) = −Zu
0Zu
z
2zν(dx)dz
=Zu
0
2z(ν(z)−ν(u))dz=Zu
0
2zν(z)dz−u2ν(u)∼α
2−αu2ν(u).
Since X∈ Zα,ρ, [19, Thm 2] implies that g−1(u)u−2σ2(u)→c0for some c0>0 as u↓0. Thus,
0<inf
z∈(0,1]
g−1(z)
z2σ2(z)≤inf
u,t∈(0,1]
g−1(ug(t))
u2g(t)2σ2(ug(t)).
Since gis regluarly varying with index 1/α, we suppose that g(t) = t1/α$(t) for a slowly varying
function $. Thus, Potter’s bounds [9, Thm 1.5.6] imply that, for some constant c > 1 and all
t, u ∈(0,1], we have $(t)/$(tuβ)≤cu−βδ for δ= 1/β −1/α > 0. Hence, we obtain ug(t)≤cg(tuβ)
and moreover g−1(ug(t)) ≤cβtuβfor all t∈(0,1] and u∈(0,1/c]. Multiplying the rightmost term on
the display above (before taking infimum) by tuβ/g−1(ug(t)) gives
(26) inf
t∈(0,1] inf
u∈(0,1/c]uβ−2σ2
Qt(u) = inf
t∈(0,1] inf
u∈(0,1/c]
tuβ
u2g(t)2σ2(ug(t)) >0.
Hence, [28, Lem. 2.3] gives the desired result.
Part (b). As before, we see that σ2
e
Qt(u) = σ2(u√t). Hence, the left side of (26) gives
inf
t∈[1,∞)inf
u∈(0,1] uβ−2σ2
e
Qt(u) = inf
u∈(0,1] uβ−2σ2(u)>0,
for any β∈(0, α). Thus, [28, Lem. 2.3] gives the desired result. 
Lemma 4.5. Let X∈ Zα,ρ for some α∈(0,1) and ρ∈(0,1], denote by gits normalising function
and define G(t) = t/g(t)for t > 0. The following statements hold for any sequences (un)n∈N⊂(0,∞)
and (sn)n∈N⊂(0,∞)such that un→ ∞ and sn↓0as n→ ∞:
(i) if unG−1(s−1
n)→ ∞, then Ψsn(un)∼ρlog(unG−1(s−1
n)),

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 23
(ii) if unG−1(s−1
n)→0, then Ψsn(un) = O([unG−1(s−1
n)]q+sn)for any q∈(0,1] with q < 1/α −1.
Proof. Part (i). Define Qt:= (Xt−γ0t)/g(t) and note that
Ψsn(un) = Z∞
0
(1 −e−tun)e−λtP0< Qt≤snG(t)dt
t,for all n∈N.
Fix δ∈(0, ρ/3), let κn:=G−1(δ/sn) and note that κn↓0 as n→ ∞. We will now split the integral in
the previous display at κnand 1 and find the asymptotic behaviour of each of the resulting integrals.
The integral on [1,∞) is bounded as n→ ∞:
Z∞
1
(1 −e−tun)e−λtP0< Qt≤snG(t)dt
t≤Z∞
1
e−λt dt
t<∞.
Next, we consider the integral on [κn,1). By Lemma 4.4(a), there exists a uniform upper bound C > 0
on the densities of Qt,t∈(0,1]. An application of [9, Thm 1.5.11(i)] gives, as n→ ∞,
Z1
κn
(1 −e−unt)e−λtP0< Qt≤snG(t)dt
t≤CZ1
κn
snG(t)dt
t∼αC
1−αsnG(κn) = δαC
1−α<∞.
Since we will prove that Ψsn(un)→ ∞ as n→ ∞, the asymptotic behaviour of Ψsn(un) will be driven
by asymptotic behaviour of the integral on (0, κn):
(27) J0
n:=Z1
0
(1 −e−unκnt)e−λκntP0< Qκnt≤snG(κnt)dt
t.
We will show that, asymptotically as n→ ∞, we may replace the probability in the integrand with
the probability P(0 < Z < δt1−1/α) in terms of the limiting α-stable random variable Z. Since Z
has a bounded density (see, e.g. [34, Ch. 4]), the weak convergence Qt
d
−→ Zas t↓0 implies that the
distributions functions converge in Kolmogorov distance by [27, 1.8.31–32, p. 43]. Thus, since κn→0
as n→ ∞, there exists some Nδ∈Nsuch that
sup
n≥Nδ
sup
t∈(0,κn]
sup
x∈R|P(0 < Qt≤x)−P(0 < Z ≤x)|< δ,
where δ∈(0, ρ/3) is as before, arbitrary but fixed. In particular, the following inequality holds
supn≥Nδsupt∈(0,κn]|P(0 < Qt≤snG(t)) −P(0 < Z ≤snG(t))|< δ. For any N≥Nδthe triangle
inequality yields
Bδ,N := sup
n≥N
sup
t∈(0,1] |P(0 < Z < δt1−1/α)−P(0 < Qtκn≤snG(tκn))|
≤δ+ sup
n≥N
sup
t∈(0,1] |P(0 < Z < δt1−1/α)−P(0 < Z ≤snG(tκn))|
≤δ+ sup
n≥N
sup
t∈(0,1]
P(mt,n < Z < Mt,n ),
where mt,n := min{snG(tκn), δt1−1/α}and Mt,n := max{snG(tκn), δt1−1/α}. We aim to show that
Bδ,N0
δ<2δfor some N0
δ∈N.
By [34, Ch. 4], there exists K > 0 such that the stable density of Zis bounded by the function
x7→ Kx−α−1for all x > 0. Thus, since Mt,n −mt,n =|δt1−1/α −snG(tκn)|, we have
P(mt,n < Z < Mt,n )≤Km−α−1
t,n |δt1−1/α −snG(tκn)|
≤K((δt1−1/α )−α−1+ (snG(tκn))−α−1)|δt1−1/α −snG(tκn)|.
(28)
To show that this converges uniformly in t∈(0,1], we consider both summands. First, we have
(δt1−1/α )−α−1|δt1−1/α −snG(tκn)|=δ−αt1−α−(tκn)(1−α2)/α G(tκn)
κ(1−α2)/α
nG(κn),

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 24
which tends to 0 as n→ ∞ uniformly in t∈(0,1] by [9, Thm 1.5.2] since t7→ t(1−α2)/αG(t) is
regularly varying at 0 with index 1 −α > 0 (recall that gis regularly varying at 0 with index 1/α and
G(t) = t/g(t)). Similarly, since sn=δ/G(κn), we have
(snG(tκn))−α−1|δt1−1/α −snG(tκn)|=δ−α
(tκn)1−1/αG(tκn)−α−1
κ1−1/α
nG(κn)−α−1−G(tκn)−α
G(κn)−α.
Since both terms in the last line converge to δαt1−αas n→ ∞ uniformly in t∈(0,1] by [9, Thm 1.5.2],
the difference tends to 0 uniformly too. Hence, the right side of (28) converges to 0 as n→ ∞ uniformly
in t∈(0,1]. Thus, for a sufficiently large N0
δ, we have
(29) sup
n≥N0
δ
sup
t∈(0,1] |P(0 < Z < δt1−1/α)−P(0 < Qtκn≤snG(tκn))|=Bδ,N 0
δ<2δ.
We now analyse a lower bound on the integral J0
nin (27). By (29), for all n≥N0
δ, we have
J0
n≥Z1
0
(1 −e−unκnt)e−λκntP0< Z ≤δt1−1/α−2δdt
t.
Recall that κn=G−1(δ/sn), define ξn:=G−1(1/sn) and note from the regular variation of G−1that
κn/ξn→δα/(α−1) as n→ ∞, implying log(unκn)∼log(unξn) as n→ ∞ since unξn→ ∞. We split
the integral from the display above at log(unκn)−1and note that
Z1
log(unκn)−1
(1 −e−unκnt)e−λκntP0< Z ≤δt1−1/α+ 2δdt
t
≤1+2δZ1
log(unκn)−1
dt
t=1+2δlog(log(unκn)) ∼1+2δlog(log(unξn)),as n→ ∞.
For the integral over (0,log(unκn)−1), first note that, for all sufficiently large n∈N, we have
P(0 < Z ≤δt1−1/α)≥P(0 < Z ≤δlog(unκn)1/α−1)> ρ −δ, t ∈(0,log(unκn)−1),
since unκn→ ∞. Thus, we have
Zlog(unκn)−1
0
(1 −e−unκnt)e−λκntP0< Z ≤δt1−1/α−2δdt
t
≥ρ−3δe−λκn/log(unκn)Zlog(unκn)−1
0
(1 −e−unκnt)dt
t∼ρ−3δlog(unξn),as n→ ∞,
where the asymptotic equivalence follows from the fact that unκn/log(unκn)→ ∞ as n→ ∞ and
R1
0(1−e−xt)t−1dt∼log xas x→ ∞. (In fact, we have R1
0(1−e−xt)t−1dt= log x+ Γ(0, x) +γfor x > 0
where Γ(0, x) = R∞
xt−1e−tdtis the upper incomplete gamma function and γis the Euler–Mascheroni
constant.) This shows that lim infn→∞ J0
n/log(unξn)≥ρ−3δ > 0 since δ∈(0, ρ/3).
Similarly, (29) implies that for all n≥N0
δ, we have
J0
n≤Z1
0
(1 −e−unκnt)e−λκntP0< Z ≤δt1−1/α+ 2δdt
t
≤(ρ+ 2δ)Z1
0
(1 −e−unκnt)dt
t∼(ρ+ 2δ) log(unξn),as n→ ∞,
implying lim supn→∞ J0
n/log(unξn)≤ρ+ 2δ. Altogether, we deduce that
ρ−3δ≤lim inf
n→∞ Ψsn(un)/log(unξn)≤lim sup
n→∞
Ψsn(un)/log(unξn)≤ρ+ 2δ.
Since δ∈(0, ρ/3) is arbitrary and the sequence Ψsn(un)/log(unξn) does not depend on δ, we may
take δ↓0 to obtain Part (i).

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 25
Part (ii). We will bound each of the terms in Ψsn(un) = J1
n+J2
n+J3
n, where ξn=G−1(1/sn) and
J1
n:=Zξn
0
(1 −e−unt)e−λtP(0 < Qt≤snG(t)) dt
t, J2
n:=Z1
ξn
(1 −e−unt)e−λtP(0 < Qt≤snG(t)) dt
t,
and J3
n:=Z∞
1
(1 −e−unt)e−λtP(0 < Xt−γ0t≤snt)dt
t.
Recall that our assumption in part (ii) states that unξn→0 as n→ ∞. Using the elementary
inequality 1−e−x≤xfor x≥0, we obtain J1
n=O(unξn) as n→ ∞. Next we bound J3
n. Lemma 4.4(b)
shows the existence of a uniform upper bound e
C > 0 on the densities of Xt/√t. Thus it holds that
P(0 < Xt−γ0t≤snt) = P(γ0√t<Xt/√t≤(γ0+sn)√t)≤e
Csn√tand hence
J3
n≤e
CsnZ∞
1
t−1/2e−λtdt=O(sn),as n→ ∞.
It remains to bound J2
n. Let q∈(0,1] with q < 1/α −1 and C > 0 be a uniform bound on the
densities of Qt(whose existence is guaranteed by Lemma 4.4(a)). The elementary bound 1 −e−x≤xq
for x≥0 for q∈(0,1] and [9, Thm 1.5.11(i)] yield
J2
n≤Cuq
nsnZ1
ξn
tqG(t)dt
t∼C
1/α −q−1uq
nsnG(ξn)ξq
n=O(uq
nξq
n),as n→ ∞.
Proof of Theorem 2.7.Throughout this proof we let φ(u):=γu−1(log log u)r, for some γ > 0, r∈R.
Part (i). Since pis arbitrary on (1/ρ, ∞) and f(t)=1/G(tlogp(1/t)), it suffices to show that
lim supt↓0(b
C0
t+bτs−s)/f(t) = lim supt↓0Lt/f(t)<∞a.s. (Recall that Lt=C0
t+bτs−sand Ψu(w) =
−log E[e−wYu] for all u, w ≥0.) By Theorem 3.1(a), it suffices to find a positive sequence (θn)n∈N
with limn→∞ θn=∞such that P∞
n=1 exp(θntn−Ψf(tn)(θn)) <∞and lim supn→∞ f(tn)/f (tn+1 )<∞
where tn:=φ(θn).
Let θn:=enand r= 0. Note that the regular variation of fat 0 yields limsupn→∞ f(tn)/f(tn+1) =
limn→∞ f(tn)/f(tn+1) = e1−1/α . Thus, it suffices to prove that the series above is finite. Since
tn=φ(θn), it follows that tnθn=γ. Note from the definition of fthat
(30) uG−1(f(φ(u))−1) = uh(u)(log(φ(u)−1))p=γ(log(γ−1u))p∼γ(log u)p→ ∞,as u→ ∞.
By Lemma 4.5(i) we have Ψf(tn)(θn)∼ρlog(θnG−1(f(tn)−1)) as n→ ∞, since θnG−1(f(tn)−1)∼
γ(log θn)p→ ∞ as n→ ∞ by (30).
Fix some ε > 0 with (1 −ε)ρp > 1. Note that Ψf(tn)(θn)≥(1 −ε)ρp log log θnfor all sufficiently
large n. It suffices to show that the following sum is finite:
∞
X
n=1
exp γ−(1 −ε)ρp log log θn.
Since (1 −ε)ρp > 1, the sum in the display above is bounded by a multiple of P∞
n=1 n−(1−ε)ρp <∞.
Part (ii). As before, since pis arbitrary in (0,1/ρ), it suffices to show that lim supt↓0Lt/f(t)≥1
a.s. By Theorem 3.1(b), it suffices to find a positive sequence (θn)n∈Nsatisfying limn→∞ θn=∞, such
that P∞
n=1(exp(−Ψf(tn)(θn)) −exp(−θntn)) = ∞and P∞
n=1 Ψf(tn+1)(θn)<∞.
Let r=γ= 1, choose σ > 1 and ε > 0 to satisfy σ(1 + ε)ρp < 1 and set θn:=enσfor n∈N. We
start by showing that the second sum in the paragraph above is finite. Since σ > 1, (30) yields
(31) θnG−1(f(tn+1)−1)∼θn
θn+1
(log θn+1)plog log θn+1 ↓0,as n→ ∞.
Hence, Lemma 4.5(ii) with q∈(0,1] and q < 1/α −1 and (31) imply
Ψf(tn+1)(θn) = O[θnG−1(f(tn+1 )−1)]q+f(tn+1),as n→ ∞.

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 26
By (31), it is enough to show that
∞
X
n=1 θn
θn+1
(log θn+1)plog log θn+1 q
<∞,and
∞
X
n=1
f(tn+1)<∞.
Newton’s generalised binomial theorem implies that θn/θn+1 = exp(nσ−(n+ 1)σ)≤exp(−σnσ−1/2)
for all sufficiently large n. Since log θn+1 ∼nσ, we conclude that the first series in the previous display
is indeed finite. The second series is also finite since f◦his regularly varying at infinity with index
(α−1)/α < 0 (recall that tn+1 =φ(θn+1)).
Next we prove that P∞
n=1(exp(−Ψf(tn)(θn)) −exp(−θntn)) = ∞. Note that we have exp(−θntn) =
exp(−log log θn) = n−σ, which is summable. Applying Lemma 4.5(i) and (30), we see that Ψf(tn)(θn)∼
ρlog(θnG−1(f(tn)−1)) as n→ ∞. As in Part (i), it is easy to see that for every ε > 0, the inequality
Ψf(tn)(θn)≤(1 + ε)ρp log log θnholds for all sufficiently large n. Thus exp(−Ψf(tn)(θn)) ≥n−σ(1+ε)ρp
is not summable (since σ(1 + ε)ρp < 1): P∞
n=1 exp(−Ψf(tn)(θn)) = ∞, completing the proof. 
4.3. Upper and lower functions at time 0- proofs. Fix any λ > 0. Let Ys:=τ−1/s for s∈(0,∞)
and note that the mean jump measure of Ysis given by
Π(ds, dt):=t−1e−λtP(−t/Xt∈ds)dt,
implying Π((0, s],dt) = t−1e−λtP(Xt≤ −t/s)dt. Since b
C0is the right-inverse of bτ, we have the identity
b
C0
t=−1/Ltwhere Lt:= inf{s > 0 : Ys> t}. Thus, lim supt↓0|b
C0
t|f(t) equals 0 (resp. ∞) if and only
if lim inf t↓0Lt/f(t) equals ∞(resp. 0). Corollary 3.7 and Proposition 3.5 above are the ingredients in
the proof of Theorem 2.9.
Proof of Theorem 2.9.Since the conditions in Theorem 3.3 only involve integrating the mean measure
Π of Ynear the origin, we may ignore the factor e−λt in the definition of the mean measure Π
above. After substituting Π(du, dt) = t−1P(−t/Xt∈du)dtin conditions (17) and (21)–(22), we
obtain the conditions in (7)–(9). Thus, Corollary 3.7 and the identity b
Ct=−1/Ltyield the claims in
Theorem 2.9.
The following technical lemma which establishes the asymptotic behaviour of the characteristic
exponent Φ defined in (24). This result plays an important role in the proof of Theorem 2.13. We will
assume that X∈ Zα,ρ. For simplicity, by virtue of [9, Eq. (1.5.1) & Thm 1.5.4], we assume without
loss of generality that: g(t) = 1 for t≥1, gis continuous and decreasing on (0,1] and the function
G(t) = t/g(t) is continuous and increasing on (0,∞). Hence, the inverse G−1of Gis also continuous
and increasing.
Lemma 4.6. Let X∈ Zα,ρ for some α∈(1,2] and ρ∈(0,1) and assume E[X2
1]<∞and E[X1] = 0.
The following statements hold for any sequences (un)n∈N⊂(0,∞)and (sn)n∈N⊂R−such that
un→ ∞ and sn→ −∞ as n→ ∞:
(i) if unG−1(|sn|−1)→ ∞, then Φsn(un)∼(1 −ρ) log(unG−1(|sn|−1)),
(ii) if unG−1(|sn|−1)↓0, then Φsn(un) = O([unG−1(|sn|−1)](α−1)/2+|sn|−2).
Proof. Part (i). Denote Qt:=Xt/g(t) and note that, for all n∈N,
Φsn(un) = Z∞
0
(1 −e−tun)e−λtPQt≤snG(t)dt
t.
For every δ > 0 let κn:=G−1(δ/|sn|) and note that κn↓0 as n→ ∞. The integral in the previous
display is split at κnand we control the two resulting integrals.
We start with the integral on [κn,∞). For any q∈(0, α) we claim that K:= supt≥0E[|Qt|q]<∞.
Indeed, since E[X2
t]<∞,t−1/2g(t)Qtconverges weakly to a normal random variable as t→ ∞.

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 27
Applying [4, Lem. 3.1] gives supt≥1E[|Qt|q]t−q/2g(t)q<∞, and hence supt≥1E[|Qt|q]<∞since
t−1/2g(t) is bounded from below for t≥1. Similarly, [10, Lem. 4.8–4.9] imply that supt≤1E[|Qt|q]<∞,
and thus K < ∞. Markov’s inequality then yields
(32) K≥sup
n∈N
sup
t≥κn|sn|qG(t)qP(Qt≤snG(t)).
Let q0:=q(1 −1/α)>0 and note that G(t)−qis regularly varying at 0 with index −q0. By (32) we
have P(Qt≤snG(t)) ≤K|sn|−qG(t)−qfor all t≥κnand n∈N. Hence, Karamata’s theorem [9,
Thm 1.5.11] gives
Z∞
κn
(1 −e−unt)e−λtPQt≤snG(t)dt
t≤KZ∞
κn|sn|−qe−λtG(t)−qdt
t
∼K
q0|sn|−qG(κn)−q=K
q0δq<∞,as n→ ∞.
Thus, the integral R∞
κn(1 −e−unt)e−λtPQt≤snG(t)t−1dtis bounded as n→ ∞.
It remains to establish the asymptotic growth of the corresponding integral on (0, κn). Since the
limiting α-stable random variable Zhas a bounded density (see, e.g. [34, Ch. 4]), the weak convergence
of Qt
d
−→ Zas t↓0 extends to convergence in Kolmogorov distance by [27, 1.8.31–32, p. 43]. Thus,
there exists some Nδ∈Nsuch that
sup
n≥Nδ
sup
t∈[0,κn]|P(Qt≤snG(t)) −P(Z≤snG(t))|< δ.
Since G(κn) = δ/|sn|and P(Z≤0) = 1 −ρ, the triangle inequality yields
Bδ:= sup
n≥Nδ
sup
t∈[0,κn]|1−ρ−P(Qt≤snG(t))|≤|1−ρ−P(Z≤ −δ)|+δ.
which tends to 0 as δ↓0.
Define ξn:=G−1(1/|sn|) for and note from the regular variation of G−1that κn/ξn→δα/(α−1) as
n→ ∞, implying log(unκn)∼log(unξn) as n→ ∞ since unξn→ ∞. As in the proof of Lemma 4.5
above, we have R1
0(1 −e−xt)t−1dt∼log xas x→ ∞. Since unξn→ ∞ and ξn↓0 as n→ ∞, we have
Zκn
0
(1 −e−unt)e−λtPQt≤snG(t)dt
t≤(1 −ρ+Bδ)Zκn
0
(1 −e−unt)e−λt dt
t
∼(1 −ρ+Bδ) log(unξn),as n→ ∞.
This implies that lim supn→∞ Φsn(un)/log(unξn)≤1−ρ+Bδ. A similar argument can be used to
obtain lim inf n→∞ Φsn(un)/log(unξn)≥1−ρ−Bδ. Since δ > 0 is arbitrary and Bδ↓0 as δ↓0, we
deduce that Φsn(un)∼(1 −ρ) log(unξn) as n→ ∞.
Part (ii). We will bound each of the terms in Φsn(un) = J1
n+J2
n+J3
n, where ξn=G−1(1/|sn|) and
J1
n:=Zξn
0
(1 −e−unt)e−λtP(Xt≤snt)dt
t, J2
n:=Z∞
1
(1 −e−unt)e−λtP(Xt≤snt)dt
t,
and J3
n:=Z1
ξn
(1 −e−unt)e−λtP(Qt≤snG(t)) dt
t.
The elementary inequality 1 −e−x≤xfor x≥0 implies that the integrand of J1
nis bounded by un.
Hence, we have J1
n=O(unξn) = O((unξn)(α−1)/2) as n→ ∞.
To bound J2
n, we use Markov’s inequality as follows: since E[X2
t] = E[X2
1]tfor all t > 0, we have
P(Xt≤snt)≤E[X2
1]t/(|sn|2t2) = E[X2
1]|sn|−2t−1, for all n∈N,t > 0. Thus, we get
J2
n≤E[X2
1]
|sn|2Z∞
1
dt
t2=E[X2
1]
|sn|2=O(|sn|−2),as n→ ∞.

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 28
It remains to bound J3
n. Let r:= (α−1)/2, pick any q∈(α/2, α) and recall from Part (i) that
K= supt≥0E[|Qt|q]<∞. Note that q0=q(1 −1/α)> r, so Karamata’s theorem [9, Thm 1.5.11], the
inequality in (32) and the elementary bound 1 −e−x≤xrfor x≥0 yield
J3
n≤Kur
nZ1
ξn
tr|sn|−qG(t)−qdt
t∼Kur
n
q0−rξr
n|sn|−qG(ξn)−q=K
q0−r(unξn)r,as n→ ∞.
We conclude that J3
n=O((unξn)r) as n→ ∞, completing the proof. 
Proof of Theorem 2.13.Throughout this proof we let φ(u):=γu−1(log log u)r, for some γ > 0 and
r∈R. By Remark 2.1 we may and do assume without loss of generality that (Xt)t≥0has a finite
second moment and zero mean.
Part (i). Since pis arbitrary on (1/(1 −ρ),∞), it suffices to show that lim inft↓0|b
C0
t|f(t)>0 a.s.
where f(t) = G(tlogp(1/t)). Since b
C0
t=−1/Lt, this is equivalent to limsupt↓0Lt/f(t)<∞a.s. Recall
that Ψu(w) = log E[e−wYu] = log E[e−wbτ−1/u ] = Φ−1/u(w) for all u > 0 and w≥0. By virtue of Theo-
rem 3.1(a), it suffices to show that P∞
n=1 exp(θntn−Ψf(tn)(θn)) <∞and lim supn→∞ f(tn)/f (tn+1)<
∞for tn:=φ(θn) and a positive sequence (θn)n∈Nwith limn→∞ θn=∞.
Let θn:=enand r= 0. Note that the regular variation of fat 0 yields limsupn→∞ f(tn)/f(tn+1) =
limn→∞ f(tn)/f(tn+1) = e1−1/α . Thus, it suffices to prove that the series is finite. Since tn=φ(θn),
it follows that tnθn=γ. Note from the definition of fthat
(33) uG−1(f(φ(u))) = uφ(u)(log(φ(u)−1))p=γ(log(γ−1u))p∼γ(log u)p→ ∞,as u→ ∞.
By Lemma 4.6(i) we have Ψf(tn)(θn) = Φ−1/f (tn)(θn)∼(1 −ρ) log(θnG−1(f(tn))) as n→ ∞, since
θnG−1(f(tn)) ∼γ(log θn)p→ ∞ as n→ ∞ by (33).
Fix some ε > 0 with (1 −ε)(1 −ρ)p > 1. Note that we have Ψf(tn)(θn)≥(1 −ε)(1 −ρ)plog log θn
for all sufficiently large n. It is enough to show that the following sum is finite:
∞
X
n=1
exp γ−(1 −ε)(1 −ρ)plog log θn.
Since (1 −ε)(1 −ρ)p > 1, this sum is bounded by a multiple of P∞
n=1 n−(1−ε)(1−ρ)p<∞.
Part (ii). As before, since pis arbitrary in (0,1/(1 −ρ)), it suffices to show lim inft↓0|b
C0
t|f(t)<∞
a.s. By Theorem 3.1(b), it suffices to show that there exists some r > 0 and a positive sequence
(θn)n∈Nsatisfying limn→∞ θn=∞, such that P∞
n=1(exp(−Ψf(tn)(θn)) −exp(−θntn)) = ∞and
P∞
n=1 Ψf(tn+1)(θn)<∞.
Let γ=r= 1, choose σ > 1 and ε > 0 satisfying σ(1 + ε)p(1 −ρ)<1 (recall p(1 −ρ)<1) and set
θn:=enσ. We start by showing that the second sum is finite. Since σ > 1, (33) yields
(34) θnG−1(f(tn+1)) ∼θn
θn+1
(log θn+1)p↓0,as n→ ∞.
Hence, the time-change b
C0
t=−1/Lt, Lemma 4.6(ii) and (34) imply
Ψf(tn+1)(θn) = Φ−1/f (tn+1 )(θn) = O[θnG−1(f(tn+1))](α−1)/2+f(tn+1)2,as n→ ∞.
By (34), it is enough to show that
∞
X
n=1 θn
θn+1
(log θn+1)plog log θn+1 (α−1)/2
<∞,and
∞
X
n=1
f(tn+1)2<∞.
Newton’s generalised binomial theorem implies that θn/θn+1 = exp(nσ−(n+ 1)σ)≤exp(−σnσ−1/2)
for all sufficiently large n. Since log θn+1 ∼nσ, we conclude that the first series in the previous display

HOW SMOOTH CAN THE CONVEX HULL OF A L´
EVY PATH BE? 29
is indeed finite. The second series is also finite since f◦his regularly varying at infinity with index
−(α−1)/α (recall that tn+1 =φ(θn+1)).
Next we prove that P∞
n=1(exp(−Ψf(tn)(θn)) −exp(−θntn)) = ∞. First observe that the terms
exp(−θntn) = exp(−log log θn) = n−σare summable. Applying Lemma 4.6(i) and (33), we obtain
Ψf(tn)(θn)∼(1 −ρ) log(θnG−1(f(tn))) as n→ ∞. As in Part (i), for all sufficiently large nwe
have Ψf(tn)(θn)≤(1 + ε)p(1 −ρ) log log θn. Thus exp(−Ψf(tn)(θn)) ≥n−σ(1+ε)p(1−ρ)and, since
σ(1 + ε)p(1 −ρ)<1, we deduce that P∞
n=1 exp(−Ψf(tn)(θn)) = ∞, completing the proof. 
4.4. Proofs of Subsection 2.5.In this subsection we prove the results stated in Subsection 2.5.
Proofs of Lemmas 2.15 and 2.19.We first prove Lemma 2.15. Let s∈ L+(S) and let the function f:
[0,∞)→[0,∞) be continuous and increasing with f(0) = 0 and define the function ˜
f(t):=Rt
0f(u)du,
t≥0. Note that ms=Xτs∧Xτs−equals Cτssince τsis a contact point between t7→ Xt∧Xt−and
its convex minorant C.
Part (i). By assumption, for any M > 0 there exists δ > 0 such that C0
t+τs−s≥Mf(t) for t∈(0, δ).
Since Rt
0(C0
u+τs−s)du=Ct+τs−ms−st it follows that Ct+τs−ms−st ≥M˜
f(t) for all t∈[0, δ).
Note that the path of Xstays above its convex minorant, implying Ct+τs−ms−st ≤Xt+τs−ms−st.
Thus, Xt+τs−ms−st ≥M˜
f(t) for all t∈[0, δ), implying that lim inf t↓0(Xt+τs−ms−st)/˜
f(t)≥M.
Part (ii). Assume that ˜
fis convex on a neighborhood of 0, and that lim supt↓0(C0
t+τs−s)/f(t) = 0.
Then, for all M > 0 there exists some δ > 0 such that C0
t+τs−s≤Mf(t) for all t∈[0, δ). Integrating
this inequality gives Ct+τs−ms−st ≤M˜
f(t) for all t∈[0, δ). Since s∈ L+(S), there exists a
decreasing sequence of slopes sn↓ssuch that tn=τsn−τs↓0 and Xtn+τs∧Xtn+τs−=Ctn+τsfor all
n∈N. Thus, either Xtn+τs−ms−stn≤M˜
f(tn) i.o. or Xtn+τs−−ms−stn≤M˜
f(tn) i.o. Since ˜
f
is continuous, we deduce that lim inft↓0(Xt+τs−ms−st)/˜
f(t)≤M.
The proof of Lemma 2.19 follows along similar lines with ˜
f(t) = Rt
0f(u)−1du,t > 0, the slope
s=−∞ and m−∞ =X0= 0. 
Proof of Corollary 2.17.Part (i) follows from Theorem 2.7 and Lemma 2.15(ii).
Part (ii). Assume α∈(1/2,1). By Theorem 2.2 and Lemma 2.15(i) it suffices to prove that (2)–(4)
hold for c= 1. As described in Subsection 2.1.1, condition (6) implies (3)–(4). By Lemma 4.4, the
density of (Xt−st)/g(t) is uniformly bounded in t > 0. Hence, the following condition implies (6):
(35) Z1
0Z1
f(t/2)
1
f−1(x)dxt
g(t)dt < ∞.
Similarly, (2) holds with c= 1 if R1
0(f(t)/g(t))dt < ∞. Thus, it remains to show that (35) holds and
R1
0(f(t)/g(t))dt < ∞.
We first establish (35). Let a=α/(1 −α) and note that f(t):= 1/G(t(log t−1)p) = t1/a e$(t) where
the slowly varying function e$is given by e$(t) = logp/a(1/t)$(tlogp(1/t)). Thus, by [9, Thm 1.5.12],
the inverse f−1of fadmits the representation f−1(t) = tab$(t) for some slowly varying function b$(t).
This slowly varying function satisfies
(36) t=f−1(f(t)) = f(t)ab$(f(t)) =⇒b$(f(t)) ∼t/f (t)a∼1/e$(t)a,as t↓0.
Since a > 1, the function f−1is not integrable at 0. Thus, by Karamata’s theorem [9, Thm 1.5.11]
and (36), the inner integral in (35) satisfies
Z1
f(t/2)
1
f−1(x)dx∼1
a−1f(t/2)1−ab$(f(t))−1∼2(a−1)/a
a−1f(t)1−ae$(t)a,as t↓0.

REFERENCES 30
Since t/g(t) = t−1/a/$(t) for t > 0, condition (35) holds if and only if the following integral is finite
Z1
0
f(t)1−ae$(t)a
$(t)t−1/adt=Z1
0
logp/a(1/t)$(tlogp(1/t))
$(t)
dt
t.
The integrand is asymptotically equivalent to logp/a(1/t) since $(tlogp(1/t))/$(t)→1 as t↓0
uniformly on [0,1] by [9, Thm 2.3.1] and our assumption on $. Thus, the condition p < −amakes
the integral in display finite, proving condition (35).
To prove that R1
0(f(t)/g(t))dt < ∞, take any δ > 0 with p(1/a −δ)<−1 (recall p/a < −1 by
assumption) and apply Potter’s bound [9, Thm 1.5.6(iii)] with δto obtain, for some constant K > 0,
Z1
0
f(t)
g(t)dt=Z1
0
g(tlogp(1/t))
g(t) logp(1/t)
dt
t≤KZ1
0
logp(1/a−δ)(1/t)dt
t<∞.
Part (iii). The result follows from Corollary 2.4 and Lemma 2.15(i). 
5. Concluding remarks
The points on the boundary of the convex hull of a L´evy path where the slope increases continuously
were characterised (in terms of the law of the process) in our recent paper [2]. In this paper we address
the question of the rate of increase for the derivative of the boundary at these points in terms of
lower and upper functions, both when the tangent has finite slope and when it is vertical (i.e. of
infinite slope). Our results cover a large class of L´evy processes, presenting a comprehensive picture
of this behaviour. Our aim was not to provide the best possible result in each case and indeed many
extensions and refinements are possible. Below we list a few that arose while discussing our results in
Section 2as well as other natural questions.
•Find an explicit description of the lower (resp. upper) fluctuations in the finite (resp. infinite)
slope regime for L´evy processes in the domain of attraction of an α-stable process in terms of
the normalising function (cf. Corollaries 2.4 and 2.12). In the finite slope regime, this appears to
require a refinement of [28, Thm 4.3] for processes in this class.
•In Theorems 2.7 and 2.13 we find the correct power of the logarithmic factor, in terms of the
positivity parameter ρ, in the definition of the function ffor processes in the domain of attraction
of an α-stable process. It is natural to ask what powers of iterated logarithm arise and how
the boundary value is linked to the characteristics of the L´evy process. This question might be
tractable for α-stable processes since power series and other formulae exist for their transition
densities [34, Sec. 4], allowing higher order control of the Laplace transform Φ in Lemmas 4.5
and 4.6.
•Find the analogue of Theorems 2.7 and 2.13 for processes attracted to Cauchy process (see Re-
marks 2.8(a) and 2.14(b) for details).
•Find L´evy processes for which there exists a deterministic function fsuch that any of the following
limits is positive and finite: lim supt↓0(C0
t+τs−s)/f(t), lim inf t↓0(C0
t+τs−s)/f(t), lim supt↓0|C0
t|f(t)
or lim inf t↓0|C0
t|f(t). By Corollaries 2.4 and 2.12, such a function does not exist for the limits
lim inft↓0(C0
t+τs−s)/f(t) or lim supt↓0|C0
t|f(t) within the class of regularly varying functions and
α-stable processes with jumps of both signs.
References
[1] Frank Aurzada, Leif D¨oring, and Mladen Savov. “Small time Chung-type LIL for L´evy processes”.
Bernoulli 19.1 (2013), pp. 115–136. issn: 1350-7265. doi:10 . 3150/ 11- BEJ395.url:https :
//doi.org/10.3150/11-BEJ395.

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Acknowledgements
JGC and AM are supported by EPSRC grant EP/V009478/1 and The Alan Turing Institute under the
EPSRC grant EP/N510129/1; AM was supported by the Turing Fellowship funded by the Programme
on Data-Centric Engineering of Lloyd’s Register Foundation; DB is funded by the CDT in Mathematics
and Statistics at The University of Warwick. All three authors would like to thank the Isaac Newton
Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme for
Fractional Differential Equations where work on this paper was undertaken. This work was supported
by EPSRC grant no EP/R014604/1.
Appendix A. Elementary estimates
Recall that (γ, σ 2, ν) is the generating triplet of Xand the definition of the functions γ,σ2and ν
in (10) above.
Lemma A.1. For any p∈(0,2],t, K > 0and ε∈(0,1], the following bounds hold
E[(|Xt| ∧ K)p]≤(γ(ε)2t2+σ2(ε)t)p/2+Kpν(ε)t,
P(|Xt| ≥ K)≤(γ(ε)2t2+σ2(ε)t)/K2+ν(ε)t.
Proof. Let Xt=γ(ε)t+Jt+Mtbe the L´evy-Itˆo decomposition of Xat level ε, where Jis compound
Poisson containing all of the jumps of Xwith magnitude at least εand Mtis a martingale with
jumps of size smaller than ε. Fix t > 0 and define the event Aof not observing any jump of Jon
the time interval [0, t]. Clearly 1 −P(A) = 1 −e−ν(ε)t≤ν(ε)t. Consider the elementary inequality
|Xt|p∧Kp≤ |γ(ε)t+Mt|p1A+Kp1Ac. Taking expectations and applying Jensen’s inequality (with
the concave function x7→ xp/2on (0,∞)), we obtain the bound
E|Xt|p∧Kp≤γ(ε)2t2+EM2
tp/2+Kp(1 −P(A)),
because EMt= 0. The first inequality readily follows. The second inequality follows from the first
one: using Markov’s inequality we get
P(|Xt| ≥ K) = P(|Xt| ∧ K≥K)≤E(X2
t∧K2)/K2.
Thus, the second result follows from the first with p= 2. 

REFERENCES 34
Department of Statistics, University of Warwick, and The Alan Turing Institute, UK
Email address:david.bang@warwick.ac.uk
Email address:jorge.gonzalez-cazares@warwick.ac.uk
Email address:a.mijatovic@warwick.ac.uk