Content uploaded by Alexander Iksanov
Author content
All content in this area was uploaded by Alexander Iksanov on Aug 19, 2018
Content may be subject to copyright.
Alexander Iksanov1and Konrad Kolesko2and Matthias Meiners3
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX
PARAMETERS
Abstract. The long-term behavior of a supercritical branching random walk
can be described and analyzed with the help of Biggins’ martingales, parame-
trized by real or complex numbers. The study of these martingales with com-
plex parameters is a rather recent topic. Assuming that certain sufficient con-
ditions for the convergence of the martingales to non-degenerate limits hold, we
investigate the fluctuations of the martingales around their limits. We discover
three different regimes. First, we show that for parameters with small absolute
values, the fluctuations are Gaussian and the limit laws are scale mixtures of
the real or complex standard normal laws. We also cover the boundary of this
phase. Second, we find a region in the parameter space in which the martin-
gale fluctuations are determined by the extremal positions in the branching
random walk. Finally, there is a critical region (typically on the boundary of
the set of parameters for which the martingales converge to a non-degenerate
limit) where the fluctuations are stable-like and the limit laws are the laws
of randomly stopped L´evy processes satisfying invariance properties similar to
stability.
Keywords: branching random walk; central limit theorem; complex martin-
gales; minimal position; point processes; rate of convergence; stable processes
Subclass: MSC 60J80 ·MSC 60F15
1. Introduction
We consider a discrete-time supercritical branching random walk on the real line.
The distribution of the branching random walk is governed by a point process Zon
R. Although there are numerous papers in which Z(R) is allowed to be infinite with
positive probability, the standing assumption of the present paper is Z(R)<∞
almost surely (a. s.). At time 0, the process starts with one individual (also called
particle), the ancestor, which resides at the origin. At time 1, the ancestor dies and
simultaneously places offspring on the real line with positions given by the points of
the point process Z. The offspring of the ancestor form the first generation of the
branching random walk. At time 2, each particle of the first generation dies and has
offspring with positions relative to their parent’s position given by an independent
copy of Z. The individuals produced by the first generation particles form the
second generation of the process, and so on.
The sequence of (random) Laplace transforms of the point process of the nth
generation positions, evaluated at an appropriate λ∈Cand suitably normalized,
forms a martingale. This martingale that we denote by (Zn(λ))n∈N0, where N0:=
N∪ {0}and N:={1,2, . . .}, is called additive martingale or Biggins’ martingale.
These martingales play a key role in the study of the branching random walk, see
e.g. [7, Theorem 4] where the spread of the nth generation particles is described
in terms of additive martingales. In the same paper, Biggins showed that subject
1Taras Shevchenko National University of Kyiv, Ukraine. Email: iksan@univ.kiev.ua
2Universit¨at Innsbruck, Austria and Uniwersytet Wroc lawski, Wroc law, Poland. Email:
kolesko@math.uni.wroc.pl
3Universit¨at Innsbruck, Austria. Email: matthias.meiners@uibk.ac.at
1
2 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
to some mild conditions, Zn(λ) converges almost surely to some limit Z(λ) locally
uniformly in λfrom a certain open domain Λ ⊆C. Biggins’ result was extended
by two of the three present authors [29] to (parts of) the boundary of the set Λ.
It is natural to ask for the rate of this convergence, i.e., for the fluctuations of
Z(λ)−Zn(λ).
A partial answer to this question was given by Iksanov and Kabluchko [21], who
proved a functional central limit theorem with a random centering for Biggins’
martingale for real and sufficiently small λ. The counterpart of this result in the
context of the complex branching Brownian motion energy model has been derived
by Hartung and Klimovsky [16]. Another related statement for branching Brownian
motion can be found in the recent paper by Maillard and Pain [35], where the
fluctuations of the derivative martingale for branching Brownian motion are studied.
The authors of the present paper also investigated in [22] fluctuations of Z(λ)−
Zn(λ) for real λin the regime where the distribution of Z1(λ) belongs to the normal
domain of attraction of an α–stable law with α∈(1,2). We refer to the end of
Section 2 for a detailed account of the existing literature.
The aim of the paper at hand is to give a complete description of the fluctua-
tions of Biggins’ martingales whenever they converge while making only minimal
moment assumptions. It turns out that, apart from the Gaussian regime studied
in [21], there are two further cases. There is an extremal regime, where the fluctua-
tions are determined by the particles close to the minimal position in the branching
random walk. In this regime, the fluctuations are exponentially small with a poly-
nomial correction. And finally, there is a critical stable regime with fluctuations of
polynomial order.
2. Model description and main results
We continue with the formal definition of the branching random walk and a
review of the results on which our work is based.
2.1. Model description and known results.
The model. Set I:=Sn≥0Nn. We use the standard Ulam-Harris notation, that is,
for u= (u1, . . . , un)∈Nn, we also write u1. . . un. Further, if v= (v1, . . . , vm)∈
Nm, we write uv for (u1, . . . , un, v1, . . . , vm). For k≤n, denote u1. . . uk, the
ancestor of uin generation k, by u|k. The ancestor of the whole population is
identified with the empty tuple ∅and its position is S(∅) = 0. Let (Z(u))u∈I
be a family of i.i.d. copies of the basic reproduction point process Zdefined on
some probability space (Ω,A,P). We write Z(u) = PN(u)
j=1 Xj(u), where N(u) =
Z(u)(R), u∈ I. We assume that Z(∅) = Z. In general, we drop the argument
∅for quantities derived from Z(∅), for instance, N=N(∅). Generation 0 of the
population is given by G0:={∅}and, recursively,
Gn+1 :={uj ∈Nn+1 :u∈ Gnand 1 ≤j≤N(u)}
is generation n+1 of the process. Define the set of all individuals by G:=Sn∈N0Gn.
The position of an individual u=u1. . . un∈ Gnis
S(u):=Xu1(∅) + . . . +Xun(u1. . . un−1).
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 3
The point process of the positions of the nth generation individuals will be denoted
by Zn, that is,
Zn=X
|u|=n
δS(u)
where here and in what follows, we write |u|=nfor u∈ Gn. The sequence of point
processes (Zn)n∈N0is called a branching random walk.
We assume that (Zn)n∈N0is supercritical, i. e., E[N] = E[Z(R)] >1. Then the
generation sizes Zn(R), n∈N0form a supercritical Galton-Watson process and
thus P(S)>0 for the survival set
S:={#Gn>0 for all n∈N}={Zn(R)>0 for all n∈N}.
The Laplace transform of the intensity measure µof Zis the function
λ7→ m(λ):=ZR
e−λx µ(dx) = EX
|u|=1
e−λS(u), λ ∈C(2.1)
where λ=θ+ iηwith θ, η ∈R. (We adopt the convention from [7] and always
write θfor Re(λ) and ηfor Im(λ).) Throughout the paper, we assume that
D={λ∈C:m(λ) converges absolutely}={θ∈R:m(θ)<∞}+iRis non-empty.
For λ∈ D and n∈N0, let
Zn(λ):=1
m(λ)nZR
e−λx Zn(dx) = 1
m(λ)nX
|u|=n
e−λS(u).
Denote by Fn:=σ(Z(u) : u∈Sn−1
k=0 Nk), and let F∞:=σ(Fn:n∈N0). It is
well known and easy to check that (Zn(λ))n∈N0forms a complex-valued martingale
with respect to (Fn)n∈N0. It is called additive martingale in the branching random
walk and also Biggins’ martingale in honor of Biggins’ seminal contribution [6].
Convergence of complex martingales. Convergence of these martingales has been
investigated by various authors in the case λ=θ∈R, see e. g. [5, 6, 32]. For the
complex case, the most important sources for us are [7] and [29]. Theorem 1 of [7]
states that if
E[Z1(θ)γ]<∞for some γ∈(1,2] (B1)
and
m(pθ)
|m(λ)|p<1 for some p∈(1, γ],(B2)
then (Zn(λ))n∈N0converges a.s. and in Lpto a limit variable Z(λ). Theorem 2 in
the same source gives that this convergence is locally uniform (a. s. and in mean)
on the set Λ = Sγ∈(1,2] Λγwhere Λγ= Λ1
γ∩Λ3
γand, for γ∈(1,2],
Λ1
γ= int{λ∈ D :E[Z1(θ)γ]<∞} and Λ3
γ= int λ∈ D : inf1≤p≤γm(pθ)
|m(λ)|p<1.
In [29], convergence of the martingales (Zn(λ))n∈N0for parameters λfrom the
boundary ∂Λ is investigated. Theorem 2.1 in the cited article states that subject
to the conditions
m(αθ)
|m(λ)|α= 1 and EP|u|=1 θS(u)e−αθS(u)
|m(λ)|α≥ −log(|m(λ)|) for some α∈(1,2)
(C1)
and
E[|Z1(λ)|αlog2+
+(|Z1(λ)|)] <∞for some > 0 (C2)
4 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
with the same αas in (C1), there is convergence of Zn(λ) to some limit variable
Z(λ). The convergence holds a.s. and in Lpfor any p<α.
As has already been mentioned the fluctuations of Zn(λ) around Z(λ) as n→ ∞
are the subject of the present paper. More precisely, we find (complex) scaling
constants an=an(λ)6= 0 such that an(Z(λ)−Zn(λ)) converges in distribution to
a non-degenerate limit as n→ ∞.
2.2. Main results. We give an example before we state our main results.
Example. There are three fundamentally different regimes for the fluctuations of
Zn(λ) around its limit Z(λ). These regimes are best understood via an example,
which is in close analogy to branching Brownian motion.
Example 2.1 (Binary splitting and Gaussian increments).Consider a branching
random walk with binary splitting and independent standard Gaussian increments,
that is, Z=δX1+δX2where X1, X2are independent standard normals. Then
m(λ) = 2 exp(λ2/2) for λ∈C. For every θ∈Rand γ > 1, we have E[Z1(θ)γ]<∞.
Hence Λ = {λ∈C:m(pθ)/|m(λ)|p<1 for some p∈(1,2]}. Thus, λ∈Λ if and
only if there exists some p∈(1,2] with m(pθ)/|m(λ)|p<1. The latter inequality
is equivalent to
(1 −p)2 log 2 + p2θ2−p(θ2−η2)<0.(2.2)
It follows from the discussion in [29, Example 3.1] that λ∈Λ iff |θ| ≤ √2 log 2/2
and θ2+η2<log 2, or √2 log 2/2≤ |θ|<√2 log 2 and |η|<√2 log 2 − |θ|.
Corollary 1.1 of [21] applies to parameters θ∈Rsatisfying m(2θ)/m(θ)2<1
θ
η
Λ
√2 log 2
−1
2
i
2
Figure 1. The figure shows the different regimes of fluctuations of
Biggins’ martingales in the branching random walk with binary splitting
and independent standard Gaussian increments.
or, equivalently, |θ|<√log2. In this case, the corollary gives convergence in
distribution of (√2 exp(−θ2/2))n(Z(θ)−Zn(θ)) to a constant multiple of pZ(2θ)·X
where Xis real standard normal and independent of Z(2θ). According to [32],
Z(2θ) is non-degenerate iff |θ|<√2 log 2/2, i.e., the limit in Corollary 1.1 of [21]
is non-degenerate only for θwhich are situated on the real axis strictly between
the two red vertical lines in Figure 1. Our first result, Theorem 2.2 below, extends
Corollary 1.1 from [21] and, in this particular example, gives the convergence of
(√2 exp((λ2/2−θ2))n(Z(λ)−Zn(λ)) to a constant multiple of pZ(2θ)·Xin
the whole bounded yellow domain surrounded by red arcs and lines. Here, Xis
independent of Z(2θ) and complex standard normal if Im(λ)6= 0. For parameters λ
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 5
from the red vertical lines, the same limit relation holds, but the limit is degenerate
as Z(2θ) = 0 a. s. Indeed, 2θis then one of the two black dots in the figure.
This problem can be resolved with the help of Seneta-Heyde norming. From [2]
we know that √nZn(2θ) converges in probability to a constant multiple of the
(non-degenerate) limit D∞of the derivative martingale. Modifying the scaling in
Theorem 2.2 by the additional prefactor n1/4gives a nontrivial limit theorem where
the limit is a constant multiple of √D∞·Xwith the same Xas before which is
further independent of D∞. This is the content of Theorem 2.3 which applies to
the λfrom the vertical red lines.
A similar trick does not work for parameters from the open yellow domains
surrounded by the two triangles consisting of red vertical lines and diagonal blue
lines. There, the contribution of the minimal positions in the branching random
walk to Z(λ)−Z(λ) is too large for a limit theorem with a (randomly scaled)
normal or stable limit. Instead, it can be checked that our Theorem 2.5 applies.
The most tedious part here is to show that E[|Z(λ)|p]<∞for some suitable p,
but this can be achieved by checking that the sufficient conditions (B1) and (B2)
are fulfilled. Theorem 2.5 is based on the convergence of the point process of the
branching random walk seen from its tip [34]. The correct scaling factors provided
by the theorem are n3λ/(2ϑ)·(2 exp(λ2/2)/(4λ/ϑ))nwith ϑ=√2 log 2 and the
limit distribution has a random series representation involving the limit process of
branching random walk seen from its tip.
Finally, on the blue lines, it holds that the distribution of the martingale limit
Z(λ) is in the domain of attraction of a stable law and hence Z(λ)−Zn(λ) exhibits
stable-like fluctuations. This regime is covered by Theorem 2.9, which shows that
nλ
2αθ (Z(λ)−Zn(λ)) converges in distribution to a L´evy process independent of F∞
satisfying an invariance property similar to α-stability (the details are explained
in Example 2.7) evaluated at the limit D∞of the derivative martingale, where
α=√2 log 2/θ ∈(1,2) for θ∈(1
2√2 log 2,√2 log 2).
Weak convergence almost surely and in probability. If ζ, ζ1, ζ2, . . . are random vari-
ables taking values in C, we write
L(ζn|Fn)w
→ L(ζ|F∞) in P-probability (2.3)
(in words, the distribution of ζngiven Fnconverges weakly to the distribution of ζ
given F∞in P-probability) if for every bounded continuous function φ:C→Rit
holds that E[φ(ζn)|Fn] converges to E[φ(ζ)|F∞] in P-probability as n→ ∞. Notice
that (2.3) implies that ζnconverges to ζin distribution as n→ ∞ as for any
bounded and continuous function φ:C→Rand every strictly increasing sequence
of positive integers, we can extract a subsequence (nk)k∈Nsuch that E[φ(ζnk)|Fnk]
converges to E[φ(ζ)|F∞] a.s. Hence, by the dominated convergence theorem,
E[φ(ζnk)] = E[E[φ(ζnk)|Fnk]] →E[E[φ(ζ)|F∞]] = E[φ(ζ)] as k→ ∞.
This implies E[φ(ζn)] →E[φ(ζ)] as n→ ∞ and, therefore, ζnd
→ζ.
Analogously, we write
L(ζn|Fn)w
→ L(ζ|F∞)P-a. s. (2.4)
(in words, the distribution of ζngiven Fnconverges a. s. to the distribution of ζgiven
F∞) if for every bounded continuous function φ:C→Rit holds that E[φ(ζn)|Fn]
converges to E[φ(ζ)|F∞] a. s. as n→ ∞. Clearly, also L(ζn|Fn)w
→ L(ζ|F∞)P-a. s.
implies ζnd
→ζ.
6 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
Henceforth, we shall assume that λ∈ D satisfies θ≥0. This simplifies the
presentation of our results but is not a restriction of generality. Indeed, if θ < 0,
we may replace the point process Z=PN
j=1 δXjby PN
j=1 δ−Xjand θby −θ > 0.
Small |λ|: Gaussian fluctuations. Our first result is an extension of Corollary 1.1
in [21] to the complex case. For λ∈ D with m(λ)6= 0, we set
σ2
λ:=E[|Z1(λ)−1|2] = E[|Z1(λ)|2]−1∈[0,∞].(2.5)
Notice that σ2
θ<∞implies σ2
λ<∞since |Z1(λ)| ≤ m(θ)
|m(λ)|Z1(θ).
Throughout the paper, we call a complex random variable ζ=ξ+ iτwith
ξ= Re(ζ) and τ= Im(ζ) standard normal if ξand τare independent, identically
distributed centered normal random variables with E[|ζ|2] = E[ξ2] + E[τ2] = 1.
Theorem 2.2 (Gaussian case).Assume that λ∈ D with m(λ)6= 0 is such that
σ2
θ<∞,σ2
λ>0and m(2θ)<|m(λ)|2. Define
m=(m(2θ)if |m(2λ)|< m(2θ),
m(2λ)if |m(2λ)|=m(2θ).
Then
Lm(λ)n
mn/2(Z(λ)−Zn(λ))Fnw
→ Lσλ
√1−m(2θ)/|m(λ)|2pZ(2θ)XF∞in P-probability
(2.6)
where Xis independent of F∞. Here, Xis complex standard normal if |m(2λ)|<
m(2θ)whereas Xis real standard normal if |m(2λ)|=m(2θ).
If, additionally, either 2θ∈Λor Z(2θ) = 0 a. s., then the weak convergence in
P-probability in (2.6) can be strengthened to weak convergence P-a. s.
A perusal of the proof of Theorem 2.2 reveals that the theorem still holds when
Z(R) = ∞with positive probability, that is, our standing assumption Z(R)<∞
a. s. is not needed for this result.
Further, notice that the limit in Theorem 2.2 may vanish a.s., namely, if Z(2θ) =
0 a. s. Equivalent conditions for (Zn(2θ))n∈N0to be uniformly integrable or equiv-
alently
P(Z(2θ)>0) >0 (2.7)
are given in [32] and [5, Theorem 1.3]. For instance,
E[Z1(2θ) log+(Z1(2θ))] <∞and 2θm0(2θ)/m(2θ)<log(m(2θ)) (2.8)
imply (2.7). In particular, the condition 2θ∈Λ comfortably ensures (2.7).
However, there may be a region of λ∈Λ for which m(2θ)<|m(λ)|2and σ2
θ<∞
but Z(2θ) is degenerate at 0 as it is the case in Example 2.11. In this situation, the
assertion of Theorem 2.2 holds but the limit is degenerate at 0. This means that
2θ6∈ Λ. Typically, there is a real parameter ϑ > 0 on the boundary of Λ such that
ϑm0(ϑ)/m(ϑ) = log(m(ϑ)) (2.9)
and either 2θ=ϑor 2θ > ϑ. The second case leads to a non-Gaussian regime
in which the extremal positions dominate the fluctuations on Zn(λ) around Z(λ).
This case will be dealt with further below. In the first case, under mild moment
assumptions, a polynomial correction factor is required and a different martingale
limit figures, namely, the limit of the derivative martingale. More precisely, we
1In the example, the corresponding region is θ2+η2<log 2 and θ≥1
2√2 log 2.
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 7
suppose that (2.8) is violated because 2θ=ϑwhere ϑ > 0 is as in (2.9). Then, for
n∈N0and u∈ Gn, we define
V(u):=ϑS(u) + nlog(m(ϑ)).(2.10)
By definition and by (2.9),
EX
|u|=1
e−V(u)= 1 and EX
|u|=1
V(u)e−V(u)= 0.(2.11)
The branching random walk ((V(u))u∈Gn)n≥0is said to be in the boundary case.
Then Wn:=P|u|=ne−V(u)=Zn(ϑ)→0 a. s., but the derivative martingale
∂Wn:=X
|u|=n
e−V(u)V(u) (2.12)
converges P-a.s. under appropriate assumptions to some random variable D∞sat-
isfying D∞>0 a. s. on the survival set S, see [10] for details. Due to a result
by A¨ıd´ekon and Shi [2, Theorem 1.1], the limit D∞also appears as the limit in
probability of the rescaled martingale Wn, namely,
√nWnP
→r2
πσ2D∞(2.13)
where
σ2=EX
|u|=1
V(u)2e−V(u)∈(0,∞).(2.14)
Relation (2.13) holds subject to the conditions (2.11), (2.14) and
E[W1log2
+(W1)] <∞and E[˜
W1log+(˜
W1)] <∞(2.15)
where ˜
W1:=P|u|=1 e−V(u)V(u)+and x±:= max(±x, 0). For the case where
(2.13) holds, we have the following result.
Theorem 2.3 (Gaussian boundary case).Suppose that ϑ > 0satisfies (2.9) and
that (2.11),(2.14) and (2.15) hold for V(u) = ϑS(u) + |u|log(m(ϑ)),u∈ G.
Further, assume that λ∈ D with m(λ)6= 0 is such that σ2
θ<∞,σ2
λ>0,
m(2θ)<|m(λ)|2and 2θ=ϑ. Define
m=(m(2θ)if |m(2λ)|< m(2θ),
m(2λ)if |m(2λ)|=m(2θ)
and an:=n1/4m(λ)n
mn/2for n∈N. Then
Lan(Z(λ)−Zn(λ))Fnw
→ L√2
π
σλ
σ
√1−m(2θ)/|m(λ)|2pD∞XF∞in P-probability
(2.16)
where Xis independent of F∞. Here, Xis complex standard normal if |m(2λ)|<
m(2θ)whereas Xis real standard normal if |m(2λ)|=m(2θ).
8 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
The regime in which the extremal positions dominate. Again suppose that ϑ > 0 sat-
isfies (2.9) and that (2.11), (2.14) and (2.15) hold for V(u) = ϑS(u) +|u|log(m(ϑ)),
u∈ G. Further, assume that λ∈Λ, but 2θ > ϑ. Then, typically, Zn(2θ)→0
because (2.8) is violated because
2θm0(2θ)/m(2θ)>log(m(2θ)).
It is known, see e.g. [37], that min|u|=nV(u) is of the order 3
2log nas n→ ∞. It will
turn out that this is too slow for a result in the spirit of Theorem 2.2 in the sense
that the contributions of the particles with small positions in the nth generation
to Z(λ)−Zn(λ) are substantial, and hence no (conditionally) infinitely divisible
limit distribution can be expected. Instead, the description of the fluctuations
Z(λ)−Zn(λ) will follow from Madaule’s work [34], where the behavior of the point
processes
µn:=X
|u|=n
δVn(u)
with Vn(u):=V(u)−3
2log nwas studied. For the reader’s convenience, we state
in detail a consequence of the main result in [34].
Proposition 2.4. Suppose the branching random walk (V(u))u∈G satisfies (2.11),
(2.14) and (2.15). Further, suppose that
The branching random walk (V(u))u∈G is non-lattice. (A1)
Then there is a point process µ∞=Pk∈NδPksuch that µ∞((−∞,0]) is a.s. finite
and µnconverges in distribution to µ∞(in the space of locally finite point measures
equipped with the topology of vague convergence).
Source. This can be derived from [34, Theorem 1.1].
Let Z(1)(λ), Z (2)(λ), . . . denote independent random variables with the same dis-
tribution as Z(λ)−1 which are independent of µ∞. We consider the following series
Xext :=X
k
e−λP ∗
k
ϑZ(k)(λ) = lim
n→∞
n
X
k=1
e−λP ∗
k
ϑZ(k)(λ),(2.17)
where −∞ < P ∗
1≤P∗
2≤. . . are the atoms of µ∞arranged in increasing order.
Theorem 2.5 (Domination by extremal positions).Suppose that ϑ > 0satisfies
(2.9) and that (2.11),(2.14),(2.15) and (A1) hold for V(u) = ϑS(u)+|u|log(m(ϑ)),
u∈ G. Let λ∈Λand assume that θ∈(ϑ
2, ϑ). If there is p∈(ϑ
θ,2] satisfying
E[|Z(λ)|p]<∞, then the series Xext defined by (2.17) converges a.s. to a non-
degenerate limit. Moreover,
n3λ
2ϑm(λ)
m(ϑ)λ/ϑ n(Z(λ)−Zn(λ)) d
→Xext.
Sufficient conditions for E[|Z(λ)|p]<∞, which are easy to check, are (B1) and
(B2). Further sufficient conditions for E[|Z(λ)|p]<∞are given in Proposition 2.6
below. Finally, we should mention the upcoming paper [23] in which conditions for
the convergence in Lpfor (Zn(λ))n∈N0are provided.
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 9
The boundary of Λ: Stable fluctuations. It has been shown in [29, Theorem 2.1] that
the martingale Zn(λ) converges on a part of the boundary of Λ. More precisely,
consider the condition
m(αθ)
|m(λ)|α= 1 and θm0(θα)
|m(λ)|α= log(|m(λ)|) (C1)
and define
∂Λ(1,2) :={λ∈∂Λ∩ D : (C1) holds with α∈(1,2)}.
Theorem 2.1 in [29] says that if λ∈ D satisfies (C1) (actually, Theorem 2.1 in
[29] requires a weaker assumption) and if E[|Z1(λ)|αlog2+
+(|Z1(λ)|)] <∞for some
> 0, then (Zn(λ))n∈N0converges a. s. and in Lpfor every p < α to some limit
Z(λ) satisfying E[Z(λ)] = 1. If an additional moment assumption holds, then a
simplified version of the proof of Theorem 2.1 in [29] gives the following result.
Proposition 2.6. Suppose that λ∈ D satisfies
m(αθ)
|m(λ)|α= 1 and θm0(θα)
|m(λ)|α≤log(|m(λ)|)
for some α∈(1,2). If, additionally, E[|Z1(λ)|γ]<∞for some α < γ ≤2, then
Zn(λ)→Z(λ)in Lpfor all p<α and there exists a constant C > 0such that
P(|Z(λ)| ≥ t)≤Ct−α(2.18)
for all t > 0.
For the rest of this section, we assume that λ∈∂Λ(1,2) and that α∈(1,2)
satisfies (C1). Notice that if ϑ > 0 is defined via (2.9), then αθ =ϑin the given
situation. To determine the fluctuations of Zn(λ) around Z(λ) in this setting, we
require stronger assumptions than those of Theorem 2.1 in [29]. First of all, as
before, we define V(u) via (2.10), i.e., V(u):=ϑS(u) + nlog(m(ϑ)) for n∈N0and
u∈ Gn. Then (C1) becomes (2.11). Further, we shall require that the following
conditions hold:
E[Z1(θ)γ]<∞for some γ∈(α, 2],(2.19)
EZ1(κθ)2]<∞for some κ∈(α
2,1).(2.20)
We denote by U=U(λ) the smallest closed subgroup of the multiplicative group
C∗=C\ {0}such that
Pe−λS(u)
m(λ)∈Ufor all u∈ I= 1.
Furthermore, for simplicity of presentation, we assume that
{|z|:z∈U}=R>:= (0,∞).(2.21)
Let us now briefly describe the structure of U. If the subgroup U1=U∩ {|z|= 1}
coincides with the unit sphere {|z|= 1}, then Uis the whole multiplicative group
C∗. Otherwise, U1is a finite group and Uconsists of finitely many connected
components. By URwe denote the one-parameter subgroup of Uwhich is either
R>if U=C∗or it is the connected component of Uthat contains 1 if U6=C∗.
Clearly, URis a subgroup isomorphic to the multiplicative group R>. By γtwe
10 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
denote the canonical parametrization of URsatisfying |γt|=t. We infer that there
exists some w∈Cwith Re(w) = 1 such that
γt=tw= exp(wlog t) for all t > 0.(2.22)
Clearly, w= 1 if UR=R>. It is also worth mentioning that U1×R>'Uvia the
isomorphism T:U1×R>→U, (z, t)7→ zγt=ztw. For illustration purposes, we
interrupt the setup and discuss an example.
Example 2.7 (Binary splitting and Gaussian increments revisited).Again we con-
sider a branching random walk with binary splitting and independent standard
Gaussian increments, that is, Z=δX1+δX2with independent standard normal
random variables X1, X2. Recall that, in this situation, we have m(λ) = 2 exp(λ2/2)
for λ∈C. The parameter region we are interested in is √2 log 2/2< θ < √2 log 2
and η=√2 log 2 −θ, see Figure 1. For λ=θ+ iηfrom this region, we have
m(λ) = 2 exp((θ+ iη)2/2) = exp(p2 log 2θ+ iθη).
Therefore, Uis generated by the set
{eθ(x−√2 log 2)+iη(x−θ):x∈R}
which we may rewrite as
{eiη2·e(θ+iη)x:x∈R}.
In particular, (2.21) holds. Moreover, U1is the closed (multiplicative) subgroup
of the unit circle generated by eiη2. This group is finite if and only if 1
2πη2∈Q,
and U1={z∈C:|z|= 1}, otherwise. As θvaries over (√2 log 2/2,√2 log 2), the
square of the imaginary part, η2, ranges over the whole interval (0,1
2log 2). Thus,
for all but countably many θ, the group Uequals C∗, but for countably many θ,
Uwill consist of a finite family of ‘snails’ as depicted in the figure below. Finally,
1R
iR
Figure 2. The group Uin the case θ=√2 log 2 −pπ
10 and η=pπ
10 .
whenever U=C∗, the scaling exponent can be chosen as w= 1. When U6=C∗,
then U1is finite and the connected component of U1containing 1 is
{e(θ+iη)x:x∈R}={eλx :x∈R}={tλ/θ :t > 0}
so that w=λ/θ in this case.
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 11
By %we denote the Haar measure on Usatisfying the normalization condition
%({z∈C: 1 ≤ |z|< e}) = 1,(2.23)
i.e., %is the image of the measure %1×dt
t, where %1is the uniform distribution on
U1, via the isomorphism T.
To understand the fluctuations of Z(λ)−Zn(λ), one needs to know the tail
behavior of Z(λ). The following theorem, which is interesting in its own right,
provides the information required. For its formulation, we introduce some addi-
tional notation. We write ˆ
C=C∪ {∞} for the one-point (Alexandroff) compact-
ification of C. Further, we denote by C2
c(ˆ
C\ {0}) the set of real-valued, twice
continuously partially differentiable functions on ˆ
C\ {0}with compact support.
Finally, we remind the reader that a measure νon Cis called a L´evy measure if
ν({0}) = 0 and RC(|z|2∧1) ν(dz)<∞. A L´evy measure νis called (U, α)-invariant
if ν(uB) = |u|−αν(B) for all u∈Uand all Borel sets B⊆C\ {0}. The L´evy
measure νis called non-zero if ν(B)>0 for some Borel set Bas above.
Theorem 2.8. Let λ∈ D satisfy (C1) with α∈(1,2). Further, suppose that (2.19),
(2.20) and (2.21) hold. Then there is a non-zero (U, α)-invariant L´evy measure ν
on Csuch that
lim
|z|→0,
z∈U|z|−αE[φ(zZ(λ))] = Rφdν
for all φ∈C2
c(ˆ
C\ {0}).
We denote by (Xt)t≥0a complex-valued L´evy process which is independent of
F∞and has characteristic exponent
Ψ(x) = Zeihx,zi−1−ihx, ziν(dz), x ∈C.
Notice that Ψ is well-defined as integration by parts gives
Z{|z|≥1}
(|z| − 1) ν(dz) = Z∞
1
ν({|z| ≥ t}) dt=ν({|z| ≥ 1})Z∞
1
t−αdt<∞.
Therefore, (Xt)t≥0is the L´evy process associated with the L´evy-Khintchine char-
acteristics (0,−R{|z|>1}z ν(dz), ν ) (cf. [27, p. 291, Corollary 15.8]).
Now we are ready to describe the fluctuations of Z(λ)−Zn(λ) for λ∈∂Λ(1,2).
Theorem 2.9. Suppose that the assumptions of Theorem 2.8 hold. Then there
exists w∈Csuch that Re(w)=1(see (2.22) for the definition of w) and
Lnw
2α(Z(λ)−Zn(λ)) | Fnw
→ L(XcD∞|F∞)in P-probability (2.24)
for c:=q2
πσ2,σ2defined by (2.14) with V(u)as in (2.10) and D∞being the a.s.
limit of the derivative martingale defined in (2.12).
Related literature. The martingale convergence theorem guarantees the almost sure
convergence of Zn(θ), but its limit Z(θ) may vanish a. s. Equivalent conditions
for P(Z(θ) = 0) <1 can be found in [6, Lemma 5], [32] and [5, Theorem 1.3].
Convergence in distribution of an(Z(λ)−Zn(λ)) as n→ ∞ for constants an>0
can be viewed as a result on the rate of convergence. In [3, 21, 22, 24, 25] the rate of
12 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
convergence of Zn(θ) to Z(θ) has been investigated in the regime P(Z(θ) = 0) <1.
The papers [3, 24, 25] deal with the issue of convergence of the infinite series
P∞
n=0 an(Z(θ)−Zn(θ)).(2.25)
More precisely, in [3], necessary conditions and sufficient conditions for the conver-
gence in Lpof the infinite series in (2.25) are given in the situation where an=ean
for some a > 0. Sufficient conditions for the almost sure convergence of the series
in (2.25) have been provided in the case where an=ean for some a > 0 in [24] and
in the case where (an)n∈N0is regularly varying at +∞in [25].
The papers [21, 22] are in the spirit of the article at hand. In these works,
for α∈(1,2], it is shown that if κ:=m(αθ)/m(θ)α<1, then κ−n/α (Z(θ)−
Zn(θ)) converges in distribution to a random variable Z(αθ)1/αUwhere Uis a
(non-degenerate) centered α–stable random variable (normal, if α= 2) independent
of Z(αθ). Specifically, the case where α∈(1,2) and P(Z1(θ)> x)∼cx−αfor some
c > 0 is covered in [22, Corollary 1.3] whereas the case α= 2 and E[Z1(θ)2]<
∞is investigated in [21, Corollary 1.1]. Both papers actually contain functional
versions of these convergences. The aforementioned assertions are extensions of the
corresponding results for Galton-Watson processes [13, 18, 19].
The counterpart of our Theorem 2.2, which gives the fluctuations of Biggins’
martingales for small parameters has a natural analogue in the complex branching
Brownian motion energy model. The corresponding statement in the latter model
is [16, Theorem 1.4].
It is well known that if θm0(θ)/m(θ) = log(m(θ)), then Zn(θ) converges to 0 a.s.
In this case a natural object to study is the derivative martingale (Dn)n∈N0. In
order to study the fluctuations of Dnaround its limit D∞one needs an additional
correction term of order (log n)/√n. The corresponding result, again in the context
of branching Brownian motion, is given in [35], where it is shown that √n(D∞−
Dn+log n
√2πn D∞)d
→SD∞for an independent 1–stable L´evy process (St)t≥0.
The martingale limits Z(λ) solve smoothing equations, namely,
Z(λ) = X
|u|=1
e−λS(u)
m(λ)[Z(λ)]ua. s. (2.26)
where the [Z(λ)]u,u∈Nare independent copies of Z(λ) which are independent of
the positions S(u), |u|= 1. If Uis centered α–stable and independent of Z(αθ),
then the limit variable Z(αθ)1/αUin [21, Corollary 1.1] and [22, Corollary 1.3]
satisfies
Z(αθ)1/α U=X
|v|=1
e−αθS(v)
m(αθ)[Z(λ)]v1
α
Ulaw
=X
|v|=1
e−θS(v)
m(αθ)1/α [Z(λ)]1/α
vUv
where (Uv)v∈Nis a family of independent copies of Uwhich is independent of all
other random variables appearing on the right-hand side of the latter distributional
equality. Hence, the distribution of Z(αθ)1/α Uis a solution to the following fixed-
point equation of the smoothing transformation:
Xlaw
=X
j≥1
TjXj(2.27)
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 13
where Tj:={j∈G1}e−θS(j)
m(αθ)1/α and the Xj,j∈Nare independent copies of the
random variable X. In (2.27), which should be seen as an equation for the distribu-
tion of Xrather than the random variable Xitself, T1, T2, . . . are considered given
whereas the distribution of Xis considered unknown. Equation (2.27) has been
studied in depth in the case where the Tjand Xjare nonnegative, see [4] for the
most recent contribution and an overview of earlier results. If, however, we consider
complex Z(λ) at complex parameters, (2.26) becomes an equation between complex
random variables and it is reasonable to conjecture that the limiting distributions
of an(Z(λ)−Zn(λ)) are solutions to (2.27) with complex-valued Tjand Xj. A
systematic study of (2.27) in the case where Tjand Xjare complex-valued has
been addressed only recently in [36].
3. Preliminaries
In this section, we fix some notation and set the stage for the proofs of our main
results.
3.1. Notation.
Complex numbers. Throughout the paper, we identify Cand R2. For instance, for
z∈C, we sometimes write z1for Re(z) and z2for Im(z). Further, we sometimes
identify z∈Cwith the column vector (z1, z2)Tand write zTfor the row vector
(z1, z2). As usual, we write zfor the complex conjugate of z∈C, i.e., z=z1−iz2.
In some proofs, we identify a complex number z=reiϕwith the matrix rR(ϕ)
where R(ϕ) is the 2 ×2 rotation matrix
R(ϕ) = cos ϕ−sin ϕ
sin ϕcos ϕ.
By ˆ
Cwe denote the one-point compactification of C, i.e., ˆ
C=C∪ {∞} and a set
K⊆ˆ
Cis relatively compact if it is relatively compact in Cor the complement of
a bounded subset of C. A function φ:ˆ
C→Ris differentiable at ∞if ψ:C→R
with ψ(z) = φ(1/z) for z6= 0 and ψ(0) = φ(∞) is differentiable at 0.
Conditional expectations. Throughout the paper, we write Pn(·) for P(·|Fn) for ev-
ery n∈N0. The corresponding (conditional) expectation and variance are denoted
En[·]:=E[·|Fn] and Varn[·]:=Var[·|Fn]. We further write E[X;A] for E[XA],
Var[X;A] for Var[XA], and Cov[X;A] for the covariance matrix of the vector
XA. If Xis a complex random variable, we write Cov[X] for the covariance ma-
trix of the vector (Re(X),Im(X))T. We also use the analogous notation with E,
Var and Cov replaced by En,Varnand Covn.
The martingale. Further, when λ∈Λ is fixed, we sometimes write Znfor Zn(λ)
and Zfor Z(λ) in order to unburden the notation.
3.2. Background and relevant results from the literature.
14 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
Recursive decomposition of tail martingales. Throughout the paper, we denote by
[·]u,u∈ I the canonical shift operators, that is, for any function Ψ of (Z(v))v∈I,
we write [Ψ]ufor the same function applied to the family (Z(uv))v∈I. Using this
notation, we obtain the following decomposition of Z(λ)−Zn(λ):
Z(λ)−Zn(λ) = m(λ)−nX
|u|=n
e−λS(u)([Z(λ)]u−1) a. s., (3.1)
which is valid for every n∈N0. Therefore, with respect to Pn,Z(λ)−Zn(λ) is a
sum of i.i.d. centered random variables. This explains the appearance of (randomly
scaled) normal or stable distributions in our main theorems.
Minimal position: First order. If θ > 0 with m(θ)<∞, then [9, Theorem 3] gives
sup
|u|=n
e−θS(u)
m(θ)n→0 a. s. as n→ ∞.(3.2)
4. The Gaussian regime
Before we prove Theorems 2.2 and 2.3, we recall some basic facts about complex
random variables.
Covariance calculations. The proofs of Theorems 2.2 and 2.3 are based on covari-
ance calculations for complex random variables. We remind the reader of some
simple but useful facts in this context. If ζ=ξ+ iτis a complex random variable
with ξ= Re(ζ) and τ= Im(ζ), then a simple calculation shows that the covariance
matrix of ζcan be represented as
Cov[ζ] = E[ξ2]E[ξτ ]
E[ξτ ]E[τ2]=1
2Re(E[|ζ|2] + E[ζ2]) Im(E[|ζ|2] + E[ζ2])
Im(E[|ζ|2] + E[ζ2]) Re(E[|ζ|2]−E[ζ2]).(4.1)
Thus covariance calculations can be reduced to second moment calculations.
Proof of Theorems 2.2 and 2.3. Throughout the paragraph, for n∈N0and u∈ Gn,
we set Yu:=e−λS(u)/mn/2. We start with a lemma.
Lemma 4.1. Suppose that λ∈ D with m(λ)6= 0 is such that σ2
θ<∞,σ2
λ>0and
m(2θ)<|m(λ)|2. then
E[|Z(λ)−1|2] = E[|Z1−1|2]
1−m(2θ)
|m(λ)|2
<∞and E[(Z(λ)−1)2] = E[(Z1−1)2]
1−m(2λ)
m(λ)2
.(4.2)
Proof. Observe that (B1) and (B2) are satisfied with γ=p= 2. Consequently,
E[|Z−1|2]<∞. In the next step, we calculate E[|Z−1|2] and E[(Z−1)2].
(Actually, the calculations below again give E[|Z−1|2]<∞.) As the increments
of square-integrable martingales are uncorrelated,
E[|Z−1|2] = lim
n→∞ E[(Zn−1)(Zn−1)] = ∞
X
n=0
E[|Zn+1 −Zn|2]
=E[|Z1−1|2]∞
X
n=0
EX
|u|=n
e−2θS(u)
|m(λ)|2n=E[|Z1−1|2]
1−m(2θ)/|m(λ)|2.
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 15
Analogously, we infer
E[(Z−1)2] = E[(Z1−1)2]∞
X
n=0
EX
|u|=n
e−2λS(u)
m(λ)2n=E[(Z1−1)2]
1−m(2λ)/m(λ)2.
Our combined proof of Theorems 2.2 and 2.3 is based on an application of the
Lindeberg-Feller central limit theorem.
Proof of Theorems 2.2 and 2.3. Recall that m=m(2θ) if |m(2λ)|< m(2θ) and
m=m(2λ) if |m(2λ)|=m(2θ). For n∈N, define cn= 1 in the situation
of Theorem 2.2 and cn:=n1/4in the situation of Theorem 2.3. Further, let
an:=cnm(λ)n
mn/2for n∈N. Then (3.1) can be rewritten in the form
an(Z−Zn) = cnX
|u|=n
Yu([Z]u−1).(4.3)
The right-hand side of (4.3) given Fnis the sum of independent centered random
variables. We show that the distribution of this sum given Fnconverges in proba-
bility to the distribution of a complex or real normal random variable. To this end,
we check the Lindeberg-Feller condition. For any ε > 0, using that |m|=m(2θ),
we obtain
X
|u|=n
En[|cnYu([Z]u−1)|2{|cnYu([Z]u−1)|2>ε}] = c2
nX
|u|=n|Yu|2σ2
λ(εc−2
n|Yu|−2)
=c2
nX
|u|=n
e−2θS(u)
m(2θ)nσ2
λ(εc−2
n|Yu|−2)
where, for x≥0,
σ2
λ(x):=E[|Z−1|2{|Z−1|2>x}].
By Lemma 4.1, we have E[|Z−1|2]<∞. The dominated convergence theorem
thus yields σ2
λ(x)↓0 as x↑ ∞. Moreover, in the situation of Theorem 2.2,
c2
nsup
|u|=n|Yu|2= sup
|u|=n
e−2θS(u)
m(2θ)n→0 a. s. as n→ ∞
by (3.2) (applied with θreplaced by 2θ). In the situation of Theorem 2.3,
c2
nsup
|u|=n|Yu|2=n1/2sup
|u|=n
e−V(u)→0 in P-probability as n→ ∞
by Proposition A.3. In any case, we conclude that
X
|u|=n
En[|cnYu([Z]u−1)|2{|cnYu([Z]u−1)|2>ε}]
≤c2
nZn(2θ)σ2
λ(ε(cnsup|u|=n|Yu|)−2)→0
as n→ ∞ a. s. or in P-probability, respectively, having utilized (2.13) for the con-
vergence in P-probability. By (4.1), covariance calculations can be reduced to calcu-
lations for the second absolute (conditional) moment and the second (conditional)
16 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
moment of m(λ)n
mn/2(Z−Zn):
En|an(Z−Zn)|2=c2
nEn X
|u|=n
Yu([Z]u−1) X
|v|=n
Yv([Z]v−1)
=c2
nEnX
|u|=n|Yu|2|[Z]u−1|2=E[|Z−1|2]c2
nX
|u|=n|Yu|2
=E[|Z−1|2]c2
nZn(2θ).(4.4)
where the second equation follows from the fact that, for |u|=|v|=nwith u6=v,
[Z]u−1 and [Z]v−1 are independent and centered, and hence the cross terms vanish.
The right-hand side of (4.4) converges to E[|Z−1|2]Z(2θ) a. s. in the situation of
Theorem 2.2 and to E[|Z−1|2]( 2
πσ2)1/2D∞in P-probability in the situation of
Theorem 2.3. An analogous calculation gives
En(an(Z−Zn))2] = E[(Z−1)2]c2
nX
|u|=n
Y2
u.(4.5)
We shall find the limit of the right-hand side of (4.5), thereby verifying that the
conditions [17, Eqs. (2.5)–(2.7)] are fulfilled. The claimed convergence then follows
from the cited source and the Cram´er-Wold device [27, p. 87, Corollary 5.5]. In the
situation of Theorem 2.2, if Zn(2θ)→0 a. s., then P|u|=nY2
u→0 a. s., so that
nothing remains to be shown. Thus, for the remainder of the proof, we suppose
that Zn(2θ) converges a.s. and in L1to Z(2θ) or that (2.13) holds. We distinguish
two cases.
Case 1 : Let |m(2λ)|< m(2θ). We apply Lemma A.4 with (L(u))u∈G = (Y2
u)u∈G.
In this case
EX
|u|=1 |L(u)|=EX
|u|=1 |Yu|2=E[Z1(2θ)] = 1.
Further,
a:=EX
|u|=1
L(u)=EX
|u|=1
Y2
u=m(2λ)
m(2θ)
satisfies |a|<1. When the assumptions of Theorem 2.2 hold, Lemma A.4(b)
applies (with condition (i) satisfied) and yields P|u|=nY2
u→0 in P-probability. If,
additionally, 2θ∈Λ, then
EX
|u|=1 |L(u)|p=EX
|u|=1 |Yu|2p=m(p2θ)
m(2θ)p<1
for some p∈(1,2]. Hence, P|u|=nY2
u→0 a. s. by Lemma A.4(a). When the
assumptions of Theorem 2.3 hold, we obtain n1/2P|u|=nY2
u→0 in P-probability
by another appeal to Lemma A.4(b) (this time with condition (ii) satisfied). Thus,
under the assumptions of both theorems, the limit of the right-hand side of (4.5)
vanishes.
Case 2 : Let |m(2λ)|=m(2θ). Then there exists some ϕ∈[0,2π) such that
m(2λ) = m(2θ)eiϕ. This implies e−2iηS(u)=eiϕfor all |u|= 1 a. s., equivalently,
S(u)∈−ϕ
2η+π
ηZfor all |u|= 1 a.s. Therefore, a. s. for every u∈ G,
e−λS(u)=e−θS (u)e−iηS(u)=±eiϕ/2e−θS(u)
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 17
and thereupon m(λ) = eiϕ/2qwhere q∈Rwith 0 <|q| ≤ m(θ). Consequently,
Zn(λ)∈Ra. s. for every n∈N0. Thus, also Z(λ)∈Ra. s. Further m(λ)n/mn/2=
m(λ)n/m(2λ)n/2=qn/m(2θ)n/2∈R. Hence, all terms in (4.4) and (4.5) coincide
and so do their limits.
It is worth noting that in Case 2, in order to arrive at the stronger statement
(weak convergence a. s.), we do not need 2θ∈Λ, but only require the uniform
integrability of (Zn(2θ))n∈N0or equivalently (2.7).
5. The regime in which the extremal positions dominate
First recall that V(u) is defined by (2.10) and that Vn(u) = V(u)−3
2log nfor
u∈ Gn. Further, for each K∈Rdefine fK:R→[0,1] by
fK(x):=
1 for x≤K,
K+ 1 −xfor K≤x≤K+ 1,
0 for x≥K+ 1.
(5.1)
Our proof of Theorem 2.5 is based on two lemmas about the processes µn,n∈N
and related point processes. Proposition 2.4 tells us that
Rfdµnd
→Rfdµ∞as n→ ∞ (5.2)
for all continuous and compactly supported f:R→[0,∞). This taken together
with information about the left tail of µnfor large nprovided by [1, Theorem 1.1]
enables us to show that relation (5.2) holds for a wider class of functions f. This
is the content of Lemma 5.1.
Lemma 5.1. Suppose that the assumptions of Theorem 2.5 are satisfied. Then
relation (5.2) holds for all continuous functions f:R→[0,∞)with f(x) = 0 for
all sufficiently large x.
Proof. Pick an arbitrary continuous function f:R→[0,∞) satisfying f(x) = 0 for
all sufficiently large x. For any fixed K∈R, the function gK(x):=f(x)(1 −fK(x))
is continuous and has a compact support. Therefore, RgKdµnd
→RgKdµ∞as
n→ ∞ by Proposition 2.4. Since µ∞((−∞, a]) <∞a. s. for any a∈Rby another
appeal to Proposition 2.4, we infer
lim
K→−∞ ZgKdµ∞=Zfdµ∞a. s. (5.3)
On the other hand, for any ε > 0,
lim sup
n→∞
PZf(x)fK(x)µn(dx)
> ε≤lim sup
n→∞
Pµn((−∞, K + 1]) ≥1
= lim sup
n→∞
Pmin
|u|=nV(u)−3
2log n≤K+ 1
where min∅:=∞. By [1, Theorem 1.1],
lim
K→−∞ lim sup
n→∞
Pmin
|u|=nV(u)−3
2log n≤K+ 1= 0.
The latter limit relation, (5.3) and [12, Theorem 4.2] imply Rfdµnd
→Rfdµ∞.
18 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
With Lemma 5.1 at hand we can now show that for any γ > 1
Re−γx µ∞(dx) = Pke−γPk<∞a. s. (5.4)
To see this, pick M > 0 and consider the following chain of inequalities:
PX
j
e−γPj> M = sup
K∈N
PX
j
e−γPjfK(Pj)> M
≤sup
K∈N
lim inf
n→∞ PX
|u|=n
e−γVn(u)fK(Vn(u)) > M
≤lim sup
n→∞
PX
|u|=n
e−γVn(u)> M
where Lemma 5.1 and the Portmanteau theorem have been used for the first in-
equality. The latter lim sup tends to 0 as M→ ∞ by [34, Proposition 2.1].
Recall that (Z(k))k∈Ndenotes a sequence of independent copies of Z(λ)−1 which
are also independent of µ∞=PkδPk. We define point processes on R×Cby
µ∗
∞:=X
k
δ(Pk,Z(k))and µ∗
n:=X
|u|=n
δ(Vn(u),[Z(λ)]u−1), n ∈N.
Lemma 5.2. Suppose that the assumptions of Theorem 2.5 are satisfied. Then
Rfdµ∗
nd
→Rfdµ∗
∞for all bounded continuous function f:R×C→Csuch that
f(x, z)=0whenever xis sufficiently large.
Proof. We derive the assertion from Lemma 5.1. More precisely, first let f:R×
C→[0,∞) be an arbitrary continuous function such that f(x, z) = 0 for all
z∈Cwhenever xis sufficiently large. Since the convergence Rfdµ∗
nd
→Rfdµ∗
∞
is equivalent to the convergence of the corresponding Laplace transforms it suffices
to show that the Laplace functional of µ∗
nat fconverges to the Laplace functional
of µ∗
∞at f. To this end, define ϕ(x):=E[exp(−f(x, Z(1) ))] for x∈R. Clearly,
0< ϕ ≤1. Further, the continuity of ftogether with the dominated convergence
theorem imply that ϕis continuous. Therefore, −log ϕ:R→[0,∞) is continuous.
Since f(x, z) = 0 for all sufficiently large x, the same is true for −log ϕ. Lemma
5.1 implies that R(−log ϕ(x)) µn(dx)d
→R(−log ϕ(x)) µ∞(dx). Using this, we find
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 19
that the Laplace functional of µ∗
nevaluated at fsatisfies
Eexp −Zf(x, y)µ∗
n(dx,dy)=EEnexp −X
|u|=n
f(Vn(u),[Z(λ)]u−1)
=EY
|u|=n
ϕ(Vn(u))
=Eexp −X
|u|=n
(−log ϕ(Vn(u)))
=Eexp −Z(−log ϕ(x)) µn(dx)
→Eexp −Z(−log ϕ(x)) µ∞(dx)
=Eexp −Zf(x, y)µ∗
∞(dx,dy).
This completes the proof for nonnegative f. For the general case, we decompose
f=f1−f2+ i(f3−f4) with fj:R×C→[0,∞) vanishing for large x. Then for
any nonnegative λj, from the first part, we conclude
Z(λ1f1+λ2f2+λ3f3+λ4f4) dµ∗
nd
→Z(λ1f1+λ2f2+λ3f3+λ4f4) dµ∗
∞
and, in particular, we infer (Rfjdµ∗
n)j=1,...,4d
→(Rfjdµ∗
∞)j=1,...,4from which we
deduce the convergence Rfdµ∗
nd
→Rfdµ∗
∞.
We now make the final preparations for the proof of Theorem 2.5. We have to
show that zn(Z(λ)−Zn(λ)) converges in distribution where
zn:=n3λ
2ϑm(λ)
m(ϑ)λ/ϑ n, n ∈N.
We shall use the decomposition
zn(Z(λ)−Zn(λ)) = zn
m(λ)nX
|u|=n
e−λS(u)([Z(λ)]u−1)
=zn
m(λ)nenλ
ϑlog m(ϑ)X
|u|=n
e−λ
ϑV(u)([Z(λ)]u−1)
=X
|u|=n
e−λ
ϑVn(u)([Z(λ)]u−1)
=X
|u|=n
e−λ
ϑVn(u)fK(Vn(u))([Z(λ)]u−1)
+X
|u|=n
e−λ
ϑVn(u)(1 −fK(Vn(u))) ·([Z(λ)]u−1)
=:Yn,K +Rn,K .
We first check that the contribution of Rn,K is negligible as Ktends to infinity.
20 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
Lemma 5.3. If the assumptions of Theorem 2.5 hold, then, for any δ > 0and
every measurable hK:R→[0,1] satisfying 0≤hK≤[K,∞)
lim
K→∞ lim sup
n→∞
PX
|u|=n
e−λ
ϑVn(u)hK(Vn(u))([Z(λ)]u−1)
> δ= 0.
Proof. Let ε, δ > 0 and 1 < β < β0:=p·θ
ϑ. From Proposition 2.1 in [34], we know
that the sequence of distributions of the random variables P|u|=ne−βVn(u),n∈N
is tight. Therefore, there is an M > 0 such that supn∈NP(Qn)≤εwhere
Qn:=nX
|u|=n
e−βVn(u)> M o.
Then
PX
|u|=n
e−λ
ϑVn(u)hK(Vn(u))([Z(λ)]u−1)
> δ
≤PX
|u|=n
e−λ
ϑVn(u)hK(Vn(u))([Z(λ)]u−1)
> δ, Qc
n+ε.
We estimate the above probability using the following strategy. First, we use
Markov’s inequality for the function x7→ |x|p. Then, given Fn, we apply Lemma
A.1. This gives
PX
|u|=n
e−λ
ϑVn(u)hK(Vn(u))([Z(λ)]u−1)
> δ, Qc
n
≤4
δp·E[|Z(λ)−1|p]·EX
|u|=n
e−β0Vn(u)hK(Vn(u))pQc
n
≤4
δp·E[|Z(λ)−1|p]·e(β−β0)KEX
|u|=n
e−βVn(u)Qc
n
≤4
δp·E[|Z(λ)−1|p]·e(β−β0)KM. (5.5)
The above bound does not depend on nand, moreover, tends to 0 as K→ ∞. The
latter is obvious since β < β0and thus limK→∞ e(β−β0)K= 0.
We conclude that
lim
K→∞ lim sup
n→∞
PX
|u|=n
e−λ
ϑVn(u)hK(Vn(u))([Z(λ)]u−1)
> δ≤ε.
The assertion follows as we may choose εarbitrarily small.
We are now ready to prove Theorem 2.5.
Proof of Theorem 2.5. Define
ˆ
Y0:= 0 and ˆ
Yn:=
n
X
k=1
e−λ
ϑP∗
kZ(k)(λ), n ∈N
and recall the notation β0=p·θ
ϑ>1. Given µ∞, for each n∈N, the random
variable ˆ
Ynis the sum of complex-valued independent centered random variables.
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 21
An application of Lemma A.1 yields
E|ˆ
Yn|pµ∞≤4E[|Z(λ)−1|p]·
n
X
k=1
e−β0P∗
k≤4E[|Z(λ)−1|p]·X
k
e−β0Pk,
and the latter sum is almost surely finite by (5.4). This shows that ( ˆ
Yn)n∈N0,
conditionally on µ∞, is an Lp-bounded martingale. We conclude that ˆ
Ynconverges
a. s. conditionally on µ∞, hence, also unconditionally thereby proving the first part
of Theorem 2.5.
The proof of the second part is based on an application of Theorem 4.2 in [12]
and the decomposition
zn(Z(λ)−Zn(λ)) = Yn,K +Rn,K .
In view of Lemma 5.3, the cited theorem gives the assertion once we have shown
the following two assertions:
1. Yn,K d
→YKas n→ ∞ for every fixed K > 0 where YKis some finite random
variable;
2. YKd
→Xext as K→ ∞.
The first assertion is a consequence of Lemma 5.2. Indeed, the function (x, z )7→
e−λ
ϑxfK(x)zis continuous and vanishes for all sufficiently large x. Therefore,
Lemma 5.2 yields
Yn,K =X
|u|=n
e−λ
ϑVn(u)fK(Vn(u))([Z(λ)]u−1)
=Ze−λ
ϑxfK(x)z µ∗
n(dx,dz)d
→Ze−λ
ϑxfK(x)z µ∗
∞(dx,dz) =:YK.
Set ˆ
Y:=Pke−λ
ϑPkZ(k)(λ) and note that ˆ
Ylaw
=Xext. To see that the second
assertion holds, we prove that E[|ˆ
Y−YK|p|µ∞]→0 a. s. as K→ ∞ which entails
YKP
→ˆ
Yas K→ ∞. To this end, we use (an infinite version of) Lemma A.1 to
obtain
E|ˆ
Y−YK|p|µ∞]≤4E[|Z(λ)−1|p·X
k
e−β0Pk1−fK(Pk))p
≤4E[|Z(λ)−1|p]·X
k
e−β0Pk{Pk>K}.
In view of (5.4) the right-hand side converges to zero a.s. as K→ ∞. The proof
of Theorem 2.5 is complete.
6. The boundary ∂Λ(1,2)
Throughout this section, we fix λ∈ D and suppose that
m(αθ)
|m(λ)|α= 1 and θm0(θα)
|m(λ)|α= log(|m(λ)|) (C1)
holds with α∈(1,2), i.e., λ∈∂Λ(1,2) and that there are γ∈(α, 2] and κ∈(α
2,1)
such that
(2.19) E[Z1(θ)γ]<∞and (2.20) EZ1(κθ)2]<∞.
As before, for n∈N0and u∈ Gn, we set L(u):=e−λS(u)/m(λ)n, and abbre-
viate Zn(λ) and Z(λ) by Znand Z, respectively. Notice that P|u|=n|L(u)|α=
22 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
P|u|=ne−V(u)for each n∈N0where V(u) is defined in (2.10), i.e., V(u):=
αθS(u) + |u|log(m(αθ)). The assumptions of Theorem 2.8 guarantee that (2.13)
holds, that is,
√nX
|u|=n|L(u)|αP
→r2
πσ2D∞.(6.1)
Indeed, (C1) and (2.19) entail conditions (2.11), (2.14) and (2.15), which are suf-
ficient for (2.13) to hold. To be more precise, (C1) implies (2.11). Further, the
function
R3t7→ mV(t) = EX
|u|=1
e−tV (u)=m(αθt)
m(αθ)t
is finite at t= 1/α < 1 since λ∈ D satisfies (C1) and at t=γ/α > 1 since, by
superadditivity,
X
|u|=1
e−θS(u)γ
≥X
|u|=1
e−γθS (u),
and (2.19) holds. Therefore, mVis finite on [1/α, γ/α] and analytic on (1/α, γ /α).
In particular, the second derivative is finite at t= 1, which yields (2.14). Again by
superadditivity, we conclude that
X
|u|=1
e−θS(u)γ
≥X
|u|=1
e−αθS(u)γ/α
.
Thus, (2.19) implies the first condition in (2.15). To see that the second condition
in (2.15) also holds, pick δ > 0 such that α−δ > 1 and use
X
|u|=1
e−θS(u)γ
≥X
|u|=1
e−(α−δ)θS(u)γ
α−δ
≥δγ
α−δ·X
|u|=1
e−αθS(u)(θS(u))+γ
α−δ
.
6.1. Martingale fluctuations on ∂Λ(1,2).First, we show how from the knowledge
of the tail behaviour of Z(λ) we can deduce Theorem 2.9. To this end, suppose
that the assumptions of Theorem 2.8 are satisfied. Set W=Z−1 and observe that
Whas the same tail behavior as Z, i.e.,
lim
|z|→0,
z∈U
E[|z|−αφ(zW )] = Rφdν(6.2)
for any φ∈C2
c(ˆ
C\{0}). To see that this is true, first notice that, for any w∈ˆ
C\{0}
and z∈Usuch that |z| ≤ 1, we have
|φ(w)−φ(w−z)|=|z|
φ(w)−φ(w−z)
z≤ |z|sup
u|∇φ(u)|P1(w),
where Pjis the j-neighborhood of supp φ, i.e., Pj={u:|u−t| ≤ jfor some t∈
supp φ}. Setting χ(w):= supu|∇φ(u)|P2∗χ0(w) where χ0:C→[0,∞) is a
probability density function smooth on Cand supported by the unit disc, we infer
that χ∈C2
c(ˆ
C\ {0}) and
|φ(w)−φ(w−z)| ≤ |z|χ(w).
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 23
Hence,
lim
|z|→0,
z∈U
E[|z|−αφ(zW )] −lim
|z|→0,
z∈U
E[|z|−αφ(zZ)]≤lim
|z|→0,
z∈U|z| · E[|z|−αχ(zZ)] = 0
where Theorem 2.8 has been used.
Our first result in this section is a corollary of Theorem 2.8. Recall that, for a
complex number z∈C, we sometimes write z1= Re(z) and z2= Im(z).
Corollary 6.1. In the situation of Theorem 2.8, for every h > 0with ν({y:|y|=
h}) = 0 and every j, k = 1,2, we have
lim
|z|→0,
z∈U|z|−αE[(zW )j(z W )k|;|zW | ≤ h] = Z{|y|<h}
yjykν(dy) (6.3)
and lim
|z|→0,
z∈U|z|−αE[zW ;|z W | ≤ h] = −Z{|y|>h}
y ν(dy).(6.4)
Proof. We start with some preparations. Throughout the proof, when letting |z| →
0 it is tacitly assumed that z∈U. First, observe that
lim sup|z|→0|z|−αE[|zW |2;|z W |< δ]→0 as δ→0.(6.5)
To see this, first choose a nonnegative function φ∈C2
c(ˆ
C\ {0}) satisfying φ≥
{|z|≥1}. Then, by (6.2),
|z|−αP(|zW |>1) ≤ |z|−αE[φ(z W )] →Rφdν
as |z| → 0. In particular, there is a finite constant C > 0 such that
sup
0<|z|≤1|z|−αP(|zW |>1) ≤C. (6.6)
In order to prove (6.5), pick δ∈(0,1). We may suppose that 0 <|z|< δ. Then
|z|−αE[|zW |2;|z W |< δ]≤ |z|−αE[(|zW | ∧ δ)2]
=|z|−αZ|z|
0
2tP(|zW |> t) dt+|z|−αZδ
|z|
2tP(|zW |> t) dt.
The first integral can be bounded above by
|z|−αZ|z|
0
2tdt=|z|2−α≤δ2−α.
Regarding the second integral, use (6.6) to arrive at
|z|−αZδ
|z|
2tP(|zW |> t) dt= 2 Zδ
|z|
t1−α|z|
t−α
P
z
tW>1dt≤2C δ2−α
2−α.
In conclusion, (6.5) holds. Further, we have to show that
lim sup|z|→0|z|−αE[|zW |;|z W |> K]→0 as K→ ∞.(6.7)
24 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
Indeed, in view of (6.6), we find
lim sup
|z|→0|z|−αE[|zW |;|z W |> K]
= lim sup
|z|→0Z∞
K
t−α
z
t
−α
P
z
tW>1dt+K1−α
z
K
−α
P
z
KW>1
≤CK 1−α1
α−1+ 1=CK 1−αα
α−1,
which tends to zero as K→ ∞. Hence, (6.7) holds.
We are ready to prove (6.4). To this end, observe that E[zW ;|z W | ≤ h] =
−E[zW ;|z W |> h] since E[W] = 0. Now pick 0 < δ < h < K such that h+δ < K
and that
ν({y:|y|=h})=0.(6.8)
Let φ∈C2
c(ˆ
C\ {0}) be of the form φ(z) = zf (|z|) with twice continuously dif-
ferentiable f: [0,∞)→[0,1] satisfying f(z) = 0 for z≤hand f(z) = 1 for
z∈[h+δ, K]. Then
lim sup
|z|→0Z{|y|>h}
y ν(dy)− |z|−αE[z W ;|zW |> h]
≤lim sup
|z|→0Zφ(y)ν(dy)− |z|−αE[φ(z W )]
+Z{h<|y|<h+δ}|y−φ(y)|ν(dy) + lim sup
|z|→0|z|−αE[|zW −φ(z W )|;h < |zW |< h +δ]
+Z{|y|>K}|y−φ(y)|ν(dy) + lim sup
|z|→0|z|−αE[|zW −φ(z W )|;|zW |> K]
≤Z{h<|y|<h+δ}|y|ν(dy) + lim sup
|z|→0|z|−α(h+δ)P(h < |zW |< h +δ)
+Z{|y|>K}|y|ν(dy) + lim sup
|z|→0|z|−αE[|zW |;|z W |> K]
having utilized (6.2) and |y−φ(y)| ≤ yfor |y| ∈ C. The first (second) term
on the right-hand side converges to zero as δ→0 in view of (6.8) (and suitable
approximation of {h<|z|<h+δ}by twice continuously differentiable functions with
subsequent application of (6.2)). The third and fourth term tend to 0 as K→ ∞
by (6.7) and since R{|y|≥1}|y|ν(dy)<∞. The latter follows from the fact that
ν({|y| ≥ t}) = t−αν({|y| ≥ 1}) which is due to the (U, α)-invariance of ν.
Turning to the proof of (6.3), we fix h > 0 satisfying (6.8) and pick j, k ∈ {1,2}.
For 0 < δ < h/2, choose f∈C2
c((0,∞)) taking values in [0,1] with f= 0 on
(0, δ/2], f= 1 on [δ, h −δ] and f= 0 on [h+δ, ∞). Define φ∈C2
c(ˆ
C\ {0})
via φ(z) = zjzkf(|z|), z∈ˆ
C. In particular, φ(z) = zjzkfor δ≤ |z| ≤ h−δand
φ(z) = 0 for |z|> h +δ. Using (6.2) with this φand (6.5) and arguing along the
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 25
lines of the proof of (6.4), we conclude that
lim sup
|z|→0|z|−αE[(z W )j(zW )k;|zW | ≤ h]−Z{|y|≤h}
yjykν(dy)
≤lim sup
|z|→0|z|−αE[|zW |2;|z W |< δ]
+ (h+δ)2lim sup
|z|→0|z|−αP(h−δ < |z W | ≤ h+δ)
+Z{|y|<δ}|y|2ν(dy)+(h+δ)2ν({h−δ < |y| ≤ h+δ}).
This bound tends to 0 as δ→0. We conclude that (6.3) holds.
We are now ready to prove Theorem 2.9.
Proof of Theorem 2.9. For any strictly increasing sequence of natural numbers, we
can pass to a subsequence (nk)k∈Nsuch that the convergence in (6.1) and (A.6)
hold a. s. along this subsequence. Once more, we use decomposition (3.1). First,
we show that the triangular array {nw/(2α)
kL(u)([Z]u−1)}|u|=nk,k∈Nis a null array.
Indeed,
sup
|u|=nk
Enk[|nw
2α
kL(u)([Z]u−1)| ∧ 1] ≤E[|Z−1|]·n1
2α
ksup
|u|=nk
e−1
αV(u)→0 a. s.
as k→ ∞ by (A.6). According to [27, Theorem 15.28 and p. 295], it suffices to
prove that, for every h > 0 with ν({z:|z|=h}) = 0,
X
|u|=nk
L(nw
2α
kL(u)[W]u| Fnk)→cD∞νvaguely in ˆ
C\ {0},(6.9)
X
|u|=nk
Covnk[nw
2α
kL(u)[W]u;|n1
2α
kL(u)[W]u| ≤ h]→cD∞Z
{|z|≤h}
zzTν(dz) a. s.,(6.10)
X
|u|=nk
Enk[nw
2α
kL(u)[W]u;|n1
2α
kL(u)[W]u| ≤ h]→ −cD∞Z
{|z|≤h}
z ν(dz) a. s. (6.11)
where c=q2
πσ2. Take any φ∈C2
c(ˆ
C\ {0}). Then, by (6.2),
lim
k→∞ X
|u|=nk
Enk[φ(nw
2α
kL(u)[W]u)] = lim
k→∞ n1/2
kX
|u|=nk
|L(u)|αRφdν=cD∞Rφdν
a. s. proving (6.9). Similarly, for (6.10) and (6.11), we can apply (6.3) and (6.4),
respectively. As a result we conclude that for any bounded continuous function
ψ:C→Rit holds that
Enk[ψ(n1/2α
k(Z−Znk))] →E[ψ(XcD∞)|F∞] a. s. (6.12)
with cas before. To summarize, we have shown that from any deterministic strictly
increasing sequence of positive integers, we can extract a deterministic subsequence
(nk)k∈Nsuch that (6.12) holds. In other words, for every bounded and continuous
ψ:C→R,
En[ψ(nw
2α(Z−Zn))] P
→E[ψ(XcD∞)|F∞] as n→ ∞,
i.e., (2.24) holds.
26 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
6.2. The tail behavior of Z(λ)for λ∈∂Λ(1,2).
An upper bound on the tails of the distribution of Z(λ)for λ∈∂Λ(1,2).
Proof of Proposition 2.6. Proposition 2.6 can be proved along the lines of the proof
of Theorem 2.1 in [29]. Equation (4.3) in the cited source carries over to the present
situation, so it suffices to show that the truncated martingale (Z(t)
n(λ))n∈N0with
increments
Z(t)
n−Z(t)
n−1=X
|u|=n−1
L(u){|L(u|j)|≤tfor j=0,...,n−1}([Z1]u−1)
satisfies
sup
n∈N0
E[|Z(t)
n−1|p]≤const ·tγ−α
where 1 ≤p<αand the constant is independent of t. (This bound is analogous
to (4.7) in [29].) To prove the above uniform bound, one may argue as in the
proof of [29, Theorem 2.1] with φ(x):=|x|γ. What is more, the fact that this
function is multiplicative and satisfies the assumptions of Lemma A.1 (Topchi˘ı-
Vatutin inequality for complex martingales), allows for a substantial simplification
of the proof given in [29, Theorem 2.1]. A combination of the uniform moment
bound above with formula (4.3) in [29] yields the desired tail bound P(|Z(λ)|>
t)≤const ·t−αfor all t > 0.
Existence of the L´evy measure ν.We now prove the following, more detailed version
of Theorem 2.8. The claim that the L´evy measure νis non-zero which is not covered
by Theorem 2.8 will be justified in the next subsection. Recall that %denotes the
Haar measure on Unormalized according to (2.23).
Theorem 6.2. Suppose that λ∈ D and that the assumptions of Theorem 2.8 are
satisfied. Then there is a (U, α)-invariant L´evy measure νon C\{0}such that for
any φ∈C2
c(ˆ
C\ {0}), we have
Zφdν= lim
|z|→0,
z∈U|z|−αE[φ(zZ)]
=−2
σ2Z|z|−αlog |z|E[φ(zZ)] −X
|u|=1
E[φ(zL(u)[Z]u)]%(dz) (6.13)
where σ2=EP|u|=1 |L(u)|α(log |L(u)|)2. Moreover,
Z|z|−αE[φ(zZ)] −X
|u|=1
E[φ(zL(u)[Z]u)]%(dz) = 0.(6.14)
For the proof of Theorem 6.2, we need the following proposition.
Proposition 6.3. Suppose that (Rn)n∈N0is a neighborhood recurrent multiplicative
random walk on Usuch that E[log |R1|] = 0 and σ2:=E[(log |R1|)2]∈(0,∞).
Further, suppose that f, h :U→Rare continuous functions satisfying |f(z)| ≤
cf(1 ∧ |z|−δ)and |h(z)| ≤ ch(|z|δ∧ |z|−δ)for some constants cf, ch, δ > 0and
f(z) = E[f(zR1)] + h(z)for all z∈U.
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 27
If there exist a sequence (zn)n∈Nin Uwith zn→0and a continuous function g
such that f(znz)→g(z)for all z∈U, then
Zh(z)%(dz)=0 and lim
|z|→0,
z∈U
f(z) = −2
σ2Zh(z) log |z|%(dz).(6.15)
Proof. From our assumptions and the dominated convergence theorem, we deduce
g(z) = lim
n→∞f(zzn) = lim
n→∞
(E[f(zznR1)] + h(zzn)) = lim
n→∞E[f(zznR1)] = E[g(zR1)].
(6.16)
Consequently, (g(zRn))n∈N0is a bounded martingale and, therefore, converges a. s.
as n→ ∞. On the other hand, (Rn)n∈N0is neighborhood recurrent on U. Using
the continuity of g, we conclude that gis constant.
Now we define stopping times τ:= inf{n∈N:|Rn|<1},T0:= 0 and, recur-
sively, Tn= inf{k≥Tn−1:|RTk| ≥ |RTn−1|} for n∈N. A variant of the duality
lemma [28, Lemma 4] then yields
EZ{|Rτ|≤|z|<1}
f(znz)%(dz)=−∞
X
k=0
EZ{|z|>|RTkzn|}
h(z)%(dz).(6.17)
The left-hand side converges to g(1)E[−log |Rτ|] as n→ ∞, and so does the right-
hand side. On the other hand, observe that the bound on himplies that it is
directly Riemann integrable (dRi) on U, cf. [14, p. 396] for the precise definition.
Moreover, for any ρ > 0, the function hρ(s):=(ρ,∞)(|s|)R{|z|>|s|} h(z)%(dz) is
also dRi on U. Hence, we infer
∞
X
k=0
EZ{|z|>|RTkzn|}
h(z)%(dz)
=∞
X
k=0
E{|RTkzn|≤ρ}Z{|z|>|RTkzn|}
h(z)%(dz)+∞
X
k=0
E[hρ(RTkzn)].(6.18)
Now suppose that c:=Rh(z)%(dz)6= 0. Then choose ρ > 0 so small that
Z{|z|>|s|}
h(z)%(dz)−c
<|c|
2
for all |s| ≤ ρ. From the renewal theorem for the group U[14, Theorem A.1], we
conclude that
∞
X
k=0
E[hρ(RTkzn)] →1
E[log |RT1|]Zhρ(s)%(ds) as n→ ∞.
On the other hand, the first infinite series in (6.18) is unbounded as n→ ∞. This
is a contradiction and, hence, Rh(z)%(dz) = 0, which is the first equality in (6.15).
From [28, Proposition 1] we infer that the function s7→ R{|z|>|s|} h(z)%(dz) is also
dRi with ZZ{|z|>|s|}
h(z)%(dz)%(ds) = Zh(z) log |z|%(dz).
An application of the renewal theorem for the group U[14, Theorem A.1] yields
that the right-hand side of (6.17) converges to
−1
E[log |RT1|]Zh(z) log |z|%(dz).
28 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
Finally, since E[log |Rτ|]·E[log |RT1|] = −σ2
2by the proof of Theorem 18.1 on p. 196
in [38], we find
g(1) = −2
σ2Zh(z) log |z|%(dz),
which does not depend on the sequence (zn)n∈N.
Further, we recall an elementary but useful fact.
Proposition 6.4. Let φ∈C2
c(ˆ
C\ {0}). Then, for any 0< ε ≤1, there is a finite
constant C > 0such that for any n∈Nand any x1, . . . , xn∈Cit holds that
φ(Pn
k=1 xk)−Pn
k=1 φ(xk)≤CP1≤j6=k≤n|xj|ε|xk|ε.
Source. The proposition which is almost identical with [15, Lemma 6.2] follows
from the proof of the cited lemma.
It is routine to check using induction on nthat the formula
E[f(Rn)] = EX
|u|=n|L(u)|αf(L(u)),(6.19)
which is assumed to hold for any bounded and measurable function f:C→R,
defines (the distribution of) a multiplicative random walk (Rn)n∈N0on Uwith i.i.d.
steps Rn/Rn−1,n∈N. From (6.19) for n= 1 and (C1), we infer E[log |R1|] = 0,
i.e., the random walk (log |Rn|)n∈N0on Rhas centered steps and thus is recurrent.
Consequently, (Rn)n∈N0is neighborhood recurrent. Moreover, by (6.19) and (2.14),
we have E[(log |R1|)2] = σ2∈(0,∞).
Theorem 6.2 will now be proved by an application of Proposition 6.3.
Proof of Theorem 6.2. For any z∈U, we define a finite measure νzon the Borel
sets of Cvia
νz(A) = |z|−αP(zZ ∈A).
First observe that, since P[|Z|> t]≤Ct−αby Proposition 2.6, the family of
measures {νz}z∈C\{0}as a subset of the set of locally finite measures on ˆ
C\ {0}is
relatively vaguely sequentially compact. Let (zn)n∈Nbe a sequence in Usatisfying
zn→0 such that νznconverges vaguely to some measure ν.
Let φ∈C2
c(ˆ
C\ {0}). Define f(z) = |z|−αE[φ(zZ )], h(z) = f(z)−E[f(zR1)]
for z∈C. We shall show that the assumptions of Proposition 6.3 are satisfied.
From the proposition we then infer that the limit limn→∞ f(zn) (hence, ν) does
not depend on the particular choice of (zn)n∈N, which implies that lim|z|→0,z∈Uf(z)
exists.
First, notice that fis continuous by the dominated convergence theorem and,
thus, also his continuous again by the dominated convergence theorem. Since φ
is bounded, we have |f(z)|≤kφk∞|z|−αwhere kφk∞:= supx∈ˆ
C\{0}|φ(x)|<∞.
Since, moreover, there is some r > 0 such that φ(z) = 0 for all |z| ≤ r, we infer
|f(z)| ≤ |z|−αkφk∞P(|zZ|> r)≤Ckφk∞r−αfor all |z|>0. Hence, |f(z)| ≤
cf·(1 ∧ |z|−α) for all z6= 0 and some cf>0. Next, we show that
|h(z)| ≤ ch(|z|δ∧ |z|−δ) (6.20)
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 29
for some ch, δ > 0 and all z6= 0. In order to prove that |h(z)| ≤ c|z|−δfor some
c > 0, it suffices to give a corresponding bound on |E[f(zR1)]|. To this end, we
first notice that for any s∈R, by (6.19), we have
E[|R1|s] = EX
|u|=1 |L(u)|α|L(u)|s=EX
|u|=1
e−λS(u)
m(λ)
α+s=m((α+s)θ)
|m(λ)|α+s.
Thus, E[|R1|s]<∞iff m((α+s)θ)<∞. By assumption m(θ)<∞and m(αθ)<
∞. Since mis convex, it is finite on the whole interval [θ, αθ]. Therefore, for
δ1∈(0, α −1), we have m((α−δ1)θ)<∞and, equivalently, E[|R1|−δ1]<∞.
Consequently,
|E[f(zR1)]| ≤ cfE[1 ∧ |zR1|−α]≤cfE[1 ∧ |zR1|−δ1]≤cfE[|R1|−δ1]· |z|−δ1
for all |z|>0. It remains to show that we may choose δ > 0 such that also
|h(z)| ≤ C|z|δ. To this end, recall that κ∈(α/2,1) is such that E[Z1(κθ)2]<∞,
see (2.20). For this κ, we obtain
|h(z)|=|f(z)−E[f(zR1)]|=|z|−α
E[φ(zZ)] −EhX
|u|=1
φ(zL(u)[Z]u)i
=|z|−α
EφX
|u|=1
zL(u)[Z]u−X
|u|=1
φ(zL(u)[Z]u)
≤C|z|2κ−αEX
u6=v|L(u)[Z]u|κ|L(v)[Z]v|κ
by Proposition 6.4. The expectation in the above expression can be estimated by
E X
|u|=1 |L(u)|κ2[E[|Z|κ]]2<∞(6.21)
where the finiteness is due to (2.20). We have shown that (6.20) holds with δ=δ1
for any δ1∈(0,(2κ−α)∧(α−1)).
Since νznconverges vaguely to ν, we have, for any φ∈C2
c(ˆ
C\ {0}),
lim
n→∞ f(zn) = lim
n→∞ Rφdνzn=Rφdν. (6.22)
Fix any z∈U. Then
lim
n→∞ f(znz) = |z|−αlim
n→∞ Rφ(zx)νzn(dx) = |z|−αRφ(zx)ν(dx) =: g(z) (6.23)
because the function t7→ φ(tz) still belongs to C2
c(ˆ
C\ {0}). Finally, we observe
that the function t7→ g(t) is continuous on ˆ
C\{0}. Consequently, Proposition 6.3
applies and shows that (6.14) holds and that
lim
|z|→0,
z∈U|z|−αE[φ(zZ)] = −2
σ2Zh(z) log |z|%(dz)
=−2
σ2Zf(z)−E[f(zR1)]log |z|%(dz)
=−2
σ2Z|z|−αlog |z|E[φ(zZ)]−EX
|u|=1
φ(zL(u)[Z]u)%(dz).
30 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
This proves (6.13). The (U, α)-invariance of νfollows from the fact that the right-
hand sides of (6.22) and (6.23) are equal for each z∈U.
The measure νis non-zero. In order to show that the L´evy measure νis non-
zero, we adopt the analytic argument invented in [14]. To this end, we take a
nondecreasing function ϕ∈C2(R>) such that ϕ(t) = 0 for t < 1/2 and ϕ(t) = 1
for t > 1. We further set ϕ(0) := 0 and ϕ(z):=ϕ(|z|) if z∈C\ {0}. For s∈C,
define κ(s) by
κ(s):=Z|z|−sEϕX
|u|=1
zL(u)[Z]u−X
|u|=1
ϕ(zL(u)[Z]u)%(dz)
whenever the absolute value of the integrand is %-integrable. Using the linearity
of the %-integral, we may write this integral as the difference of two %-integrals.
Straightforward estimates now show that both these integrals are finite if Re(s)∈
(1, α) since the latter entails m(Re(s)θ)<∞. For sin the strip 1 <Re(s)< α,
using Fubini’s theorem and the invariance of the Haar measure %, we may rewrite
κ(s) in the form
κ(s) = Z|z|−sϕ(z)%(dz)E|Z|s−X
|u|=1 |L(u)[Z]u|s
=Z|z|−sϕ(z)%(dz)·1−m(sθ)
|m(λ)|s·E[|Z|s].(6.24)
Lemma 6.5. For any s > 1there exists a finite constant Cssuch that, for any
x, y ∈C, we have
Z|z|−s|ϕ(zy)−ϕ(zx)|%(dz)≤Cs||y|s− |x|s|
and sups∈ICs<∞for every closed interval I⊂(1,∞).
Proof. Without loss of generality we assume that |x| ≤ |y|. Then |ϕ(zy)−ϕ(zx)|
may only be positive when |zy|>1/2 and |zx|<1. We shall consider the three
cases |zx|<1/2<|zy|<1, |zx|<1<|zy|and 1/2<|zx|<|zy|<1 separately.
In the second case, we conclude that the integral in focus does not exceed
Z{|x|<1
|z|<|y|} |z|−s|ϕ(zy)−ϕ(zx)|%(dz)≤Z{|x|<1
|z|<|y|} |z|−s%(dz) = 1
s(|y|s− |x|s).
Analogously, in the first case, we obtain the upper bound
Z{|x|<1
2|z|<|y|} |z|−s|ϕ(zy)−ϕ(zx)|%(dz)≤1
s2s(|y|s− |x|s).
It remains to get the bound in the third case. Since the derivative of ϕ(as a
function on R>) is bounded so that kϕ0k∞= supx>0ϕ0(x)<∞we have the upper
bound
Z
{|y|<1
|z|<2|x|}
|z|−s|ϕ(zx)−ϕ(zy)|%(dz)≤ kϕ0k∞Z
{|y|<1
|z|<2|x|}
|z|−s(|zy|−|zx|)%(dz)
=kϕ0k∞
s−1(|y|−|x|)(|2x|s−1− |y|s−1)≤kϕ0k∞
s−12s−1(|y|s− |x|s).
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 31
The latter follows from the elementary inequality
(t−1)(2s−1−ts−1)≤2s−1(t−1) ≤2s−1(ts−1),
that we use for t=|y|
|x|∈(1,2). The claim concerning the local boundedness of
s7→ Csis now obvious.
Lemma 6.6. Suppose that (2.19) and (2.20) hold. Then κis well-defined and
holomorphic on the strip 1<Re(s)< α +for some > 0.
Proof. Let s∈(1, γ) where γ∈(α, 2] is as in (2.19). First, we show that
EX
|u|=1
L(u)[Z]u
s
−max
|u|=1 |L(u)[Z]u|s<∞,(6.25)
EX
|u|=1 |L(u)[Z]u|s−max
|u|=1 |L(u)[Z]u|s<∞,and (6.26)
EZ|z|−sX
|u|=1
h(|zL(u)[Z]u|)−h(max|u|=1 |zL(u)[Z]u|)%(dz)<∞(6.27)
where h=(1,∞)or h=ϕfor ϕdefined in the paragraph preceding Lemma 6.5.
In order to prove (6.25), we first observe that
EX
|u|=1
L(u)[Z]u
s
−X
|u|=1 |L(u)[Z]u|2s/2
=EX
|u|=1|L(u)[Z]u|2+X
|u|=|v|=1
u6=v
Re(L(u)[Z]uL(v)[Z]v)s/2
−X
|u|=1|L(u)[Z]u|2s/2
≤E X
|u|=|v|=1
u6=v
|L(u)[Z]uL(v)[Z]v|s/2
=EE1 X
|u|=|v|=1
u6=v
|L(u)[Z]uL(v)[Z]v|s/2
≤(E[|Z|])s·E X
|u|=|v|=1
u6=v
|L(u)||L(v)|s/2
≤(E[|Z|])s·E X
|u|=1 |L(u)|s<∞,
where we used the subadditivity on [0,∞) of the function t7→ ts/2for the first
inequality, Jensen’s inequality for the second, and (2.19) to conclude the finiteness.
This in combination with the inequality
0≤X
|u|=1 |L(u)[Z]u|2s/2
−max
|u|=1 |L(u)[Z]u|s≤X
|u|=1 |L(u)[Z]u|s−max
|u|=1 |L(u)[Z]u|s,
32 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
which follows from the aforementioned subadditivity, shows that (6.25) is a conse-
quence of (6.26).
For h=(1,∞)or h=ϕand positive xj, we have
(1 −h(max
jxj)) ≤Y
j
(1 −h(xj)) + X
j6=k{xj>1/2, xk>1/2}∧1.
Further, there exists some finite C≥1 such that x−1 + e−x≤C(x∧x2) for all
x≥0 and P(|Z|> t)≤Ct−α,P(|Z|> t)≤C t−1for all t > 0 (the latter follows
from Markov’s inequality). Using these facts, we infer
Z|z|−sE1X
|u|=1
h(|zL(u)[Z]u|)−h(max
|u|=1 |zL(u)[Z]u|)%(dz)
≤Z|z|−sX
|u|=1
E1[h(|zL(u)[Z]u|)] −1 + e−P|u|=1 E1[h(|zL(u)[Z]u|)] %(dz)
+Z|z|−s X
|u|=|v|=1
u6=v
P1(|zL(u)[Z]u|>1/2,|zL(v)[Z]v|>1/2)∧1%(dz)
≤CZ|z|−sX
|u|=1
E1[h(|zL(u)[Z]u|)]∧X
|u|=1
E1[h(|zL(u)[Z]u|)]2
%(dz)
+ 4αC2Z|z|−s X
|u|=|v|=1
u6=v
|zL(u)|α|zL(v)|α∧1%(dz)
≤4C3Z|z|−sX
|u|=1 |L(u)||z|∧X
|u|=1 |L(u)||z|2
%(dz)
+ 4αC2X
|u|=|v|=1
u6=v
|L(u)|α|L(v)|αs/2αZ|z|−s(|z|2α∧1) %(dz)
≤4C3Z|z|−s(|z|∧|z|2)%(dz)+4αC2Z|z|−s(|z|2α∧1) %(dz)
·X
|u=1||L(u)|s
(6.28)
where in the last step we have used that (P|u|=1 |L(u)|α)s/α ≤(P|u|=1 |L(u)|)s.
Notice that the last two %-integrals in (6.28) are finite since 1 < s < 2. Assumption
(2.19) entails E[(P|u|=1 |L(u)|)s]<∞which proves (6.27). Choosing h=(1,∞)
and taking the expectation in (6.28), we conclude that (6.26) holds. In particular,
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 33
for s∈Cwith 1 < s1= Re(s)< γ, we infer
Z|z|−sE[ϕ(zZ)] −X
|u|=1
E[ϕ(zL(u)[Z]u)]
%(dz)
=Z|z|−s1
E[ϕ(zZ)] −X
|u|=1
E[ϕ(zL(u)[Z]u)]
%(dz)
≤Z|z|−s1
EϕzX
|u|=1
L(u)[Z]u−Ehϕ(max
|u|=1 |zL(u)[Z]u|)i
%(dz)
+Z|z|−s1X
|u|=1
E[ϕ(zL(u)[Z]u)] −Ehϕ(max
|u|=1 |zL(u)[Z]u|)i%(dz)
≤Cs1·EX
|u|=1
L(u)[Z]u
s1
−max
|u|=1 |L(u)[Z]u|s1
+Z|z|−s1X
|u|=1
E[ϕ(zL(u)[Z]u)] −Ehϕ(max
|u|=1 |zL(u)[Z]u|)i%(dz)<∞
by Lemma 6.5, (6.25) and (6.27). Therefore, κis well defined on the strip 1 <
Re(s)< γ. Moreover, for any closed triangle ∆ in 1 <Re(s)< γ, by the above
calculation, we can apply Fubini’s theorem and the holomorphy of s7→ |z|−sin
Re(s)>1 to conclude that
Z∂∆
κ(s) ds
=ZZ∂∆|z|sdsEϕX
|u|=1
zL(u)[Z]u−X
|u|=1
ϕ(|zL(u)[Z]u|)%(dz)=0,
which implies that κis holomorphic on the strip 1 <Re(s)< γ by Morera’s
theorem.
Theorem 6.7. If (2.19) and (2.20) hold, then the L´evy measure νis non-zero.
Proof. For s∈Cwith 0 <Re(s)< α, we define the holomorphic function F(s):=
E[|Z|s]. The functions κand Fare related by the identity, valid for 1 <Re(s)< α,
F(s) = κ(s)
1−m(sθ)
|m(λ)|s·Z|z|−sϕ(|z|)%(dz)−1
,(6.29)
which is a direct consequence of (6.24). According to Lemma 6.6, κpossesses
a holomorphic extension to some neighborhood of α. Assuming that ν= 0 we
show that Fhas such an extension as well. The latter statement will lead to a
contradiction. Indeed, if ν= 0, then Rϕ(|z|)ν(dz) = 0, which together with (6.13)
shows that κ0(α) = 0. Since also κ(α) = 0 by (6.14), we infer that the numerator
in (6.29) has a 0 of at least second order at α, while the denominator has a zero of
at most second order at αby virtue of
d2
ds2
m(sθ)
|m(λ)|s=EX
|u|=1
log2e−θS(u)
|m(λ)|e−θS (u)
|m(λ)|s>0
34 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
for all 1 < s < α +, in particular for s=α. Hence, Fdoes possess a holomorphic
extension to some neighborhood of α. We conclude from Landau’s theorem (cf.
[40, Theorems 5a and 5b in Chap. II]) that
E[|Z|α+δ]<∞,(6.30)
for some δ > 0.
By Inlet us denote an increasing family of subtrees of the Harris-Ulam tree I
such that I1={∅},|In|=nand Sn∈NIn=I. By a classical diagonal argument
such a family exists. Write unfor the unique vertex from the set In\In−1,n∈N.
Next, we define
Mn:=
n
X
k=1
L(uk)([Z1]uk−1) and M(t)
n=
n
X
k=1
L(uk){L(uk|j)≤tfor all j≤|uk|}([Z1]uk−1),
and observe that they constitute martingales with respect to the filtration (Hn)n∈N0
where Hn:=σ(Z(uk) : k= 1, . . . , n) for n∈N0. We claim that Mnconverges a. s.
to Z−1. To see this, recall from the proof of [29, Theorem 2.1] that on the set
{maxv∈G |L(v)| ≤ t}, we have
Zn−1 = Z(t)
n−1 = X
|u|≤n−1
L(u){L(u|j)≤tfor j<n}([Z1]u−1).
Further, the martingale Z(t)
n−1 converges a.s. and in Lγto some finite limit Z(t)−1
which equals Z−1 on {maxv∈G |L(v)| ≤ t}. We prove that Z(t)−1 is also the limit
in Lγof the martingale (M(t)
n)n≥0. Indeed, we write
E[|Z(t)−1−M(t)
n|γ]
=E
∞
X
k=0 X
|u|=k
L(u){L(u|j)≤tfor j≤k}([Z1]u−1)
−X
u∈In∩G
L(u){L(u|j)≤tfor j≤|u|}([Z1]u−1)
γ
=E
∞
X
k=0 X
|u|=k,
u6∈In
L(u){L(u|j)≤tfor j≤k}([Z1]u−1)
γ.
Notice that given Fkthe sum P|u|=k,u6∈InL(u){L(u|j)≤tfor j≤k}([Z1]u−1) is a
weighted sum of centered random variables and can be considered a martingale
increment. Two applications of Lemma A.1 yield
E
∞
X
k=0 X
|u|=k,
u6∈In
L(u){L(u|j)≤tfor j≤k}([Z1]u−1)
γ
≤4E∞
X
k=0 X
|u|=k,
u6∈In
L(u){L(u|j)≤tfor j≤k}([Z1]u−1)
γ
≤16 ·E[|Z1−1|γ]·E∞
X
k=0 X
|u|=k,
u6∈In
|L(u)|γ{L(u|j)≤tfor j≤k}.
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 35
The above expectations are finite by (2.19) and the arguments from the proof
of Theorem 2.1 in [29]. By the dominated convergence theorem, we infer that
limn→∞ E[|Z(t)−1−M(t)
n|γ] = 0. Since P(maxv∈G |L(v)|> t)≤t−α, we conclude
that Mn→Z−1 a. s. In view of this and (6.30),
EX
v∈G |L(v)|2|[Z1]v−1|2(α+δ)/2<∞(6.31)
by the complex version of Burkholder’s inequality. On the other hand, from [33,
Theorem 1.5] we have
P(maxu∈G |L(u)|> t)> ct−α(6.32)
for some c > 0 and all sufficiently large t. Pick t0>0 such that P(|Z1−1|> t0)≥1
2.
Denote by Ntthe set of individuals uthat are the first in their ancestral line with
the property that |L(u)|> t, i.e.,
Nt={u∈ I :|L(u)|> t, and L(u|k)≤tfor all k < |u|}.(6.33)
Then Ntis an optional line in the sense of Jagers [26, Section 4]. Denote by
FNtthe σ-field that contains the information of all reproduction point processes
of all individuals that are neither in Ntnor a descendent of a member of Nt, see
again Jagers [26, Section 4] for a precise definition. Then, by the strong Markov
branching property [26, Theorem 4.14] (the σ-field FNtwas introduced for a proper
application of this result) and (6.32), we infer
PX
u∈G |L(u)|2|[Z1]u−1|2> t2
0t2≥PX
u∈Nt|[Z1]u−1|2> t2
0
≥P(Nt6=∅)·P(|Z1−1|> t0)≥c
2t−α
for all sufficiently large t. This contradicts to (6.31), thereby proving that νis
non-zero.
Appendix A. Auxiliary results
In this section, we gather auxiliary facts needed in the proofs of our main results.
A.1. Inequalities for complex random variables. Throughout the paper, we
need the complex analogues of known inequalities for real-valued random variables.
The Topchi˘ı-Vatutin inequality for complex martingales. We begin with an exten-
sion of the Topchi˘ı-Vatutin inequality [39, Theorem 2] to complex-valued martin-
gales.
Lemma A.1. Let f: [0,∞)→[0,∞)be a nondecreasing convex function with
f(0) = 0 such that g(x):=f(√x)is concave on (0,∞). Let (Mn)n∈N0be a complex-
valued martingale with M0= 0 a. s. and set Dn:=Mn−Mn−1for n∈N. If
E[f(|Dk|)] <∞for k= 1, . . . , n, then
E[f(|Mn|)] ≤4
n
X
k=1
E[f(|Dk|)].(A.1)
Further, if f(x)>0for some x > 0and P∞
k=1 E[f(|Dk|)] <∞, then Mn→M∞
a. s. for some random variable M∞and (A.1) holds for n=∞.
36 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
Proof. We first observe that
f(|z+w|) + f(|z−w|)≤2(f(|z|) + f(|w|)) for all z, w ∈C.(A.2)
To see this, note that gis subadditive as a concave function with g(0) = 0, whence
g(x2+y2)≤g(x2) + g(y2) = f(x) + f(y) for all x, y ≥0.
Put x=|u+v|/2 and s=|u−v|/2 for u, v ∈Cand observe that
u+v
2
2+
u−v
2
2=|u|2+|v|2
2.
This together with the concavity of ggives
1
2f(|u|)+f(|v|)=1
2g(|u|2)+g(|v|2)≤g|u|2+|v|2
2≤f|u+v|
2+f|u−v|
2.
Multiply this inequality by 2 and set u=z+wand v=z−wfor z, w ∈Cto infer
(A.2).
The remainder of the proof closely follows the proof of Theorem 2 in [39]. For
k= 1, . . . , n assume that E[f(|Dk|)] <∞and denote by D∗
ka random variable such
that Dkand D∗
kare i.i.d. conditionally given Mk−1. Then
E[f(|Mk−1+Dk−D∗
k|)] = E[E[f(|Mk−1+Dk−D∗
k|)|Mk−1]]
=E[E[f(|Mk−1−(Dk−D∗
k)|)|Mk−1]]
=E[f[|Mk−1−(Dk−D∗
k)|].
An appeal to (A.2) thus yields
E[f(|Mk−1+Dk−D∗
k|)] ≤E[f(|Mk−1|)] + E[f(|Dk−D∗
k|)].(A.3)
Another application of (A.2) yields
E[f(|Dk−D∗
k|)] ≤4E[f(|Dk|)].(A.4)
Further,
|Mk−1+Dk|=|Mk−1+Dk−E[D∗
k|Mk−1, Dk]|
=|E[Mk−1+Dk−D∗
k|Mk−1, Dk]|
≤E[|Mk−1+Dk−D∗
k| | Mk−1, Dk]
by Jensen’s inequality for conditional expectation in R2, which is applicable because
the function x7→ |x|is convex on R2. Hence,
E[f(|Mk−1+Dk|)] ≤E[f(E[|Mk−1+Dk−D∗
k||Mk−1, Dk])] ≤E[f(|Mk−1+Dk−D∗
k|)]
(A.5)
by the monotonicity of fand Jensen’s inequality. Combining (A.3), (A.4) and
(A.5), we arrive at
E[f(|Mk|)] ≤Ef(|Mk−1|)+4Ef(|Dk|).
The claimed inequality follows recursively.
Finally, suppose that f(x)>0 for some x > 0 and that Pk=1 E[f(|Dk|)] <∞.
Then supn∈NE[f(|Mn|)] ≤Pk=1 E[f(|Dk|)] <∞. Since f(0) = 0 and fis convex,
we conclude that fgrows at least linearly fast and, thus, supn∈NE[|Mn|]<∞.
By the martingale convergence theorem, Mn→M∞a.s. for some random variable
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 37
M∞. Further, fis continuous as it is convex and non-decreasing. Therefore, Fatou’s
lemma implies
E[f(|M∞|)] = E[lim inf
n→∞ f(|M∞|)] ≤lim inf
n→∞ E[f(|Mn|)] ≤∞
X
k=1
E[f(|Dk|)] <∞.
Tail bounds for sums of weighted i.i.d. complex random variables. Actually, we shall
need the following corollary of Lemma A.1, which is closely related to Lemma 2.1
in [8] (see also formulae (2.3) and (2.10) in [30]) but deals with complex-valued
rather than real-valued random variables.
Corollary A.2. Let c1, . . . , cnbe complex numbers satisfying Pn
k=1 |ck|= 1. Fur-
ther, let Y1, . . . , Ynbe independent copies of a complex-valued random variable Y
with E[Y]=0and E[|Y|]<∞. Then, for ε∈(0,1) and with c:= maxk=1,...,n |ck|,
P
n
X
k=1
ckYk
> ε≤8
ε2Z1/c
0
cxP(|Y|> x) dx+Z∞
1/c
P(|Y|> x) dx.
Proof. We use Lemma A.1 with f(x) = x2[0,1](x) + (2x−1) (1,∞)(x) for x≥0.
Clearly, fis convex such that gdefined by g(x) = f(√x) = x[0,1](x) + (2x1/2−
1) (1,∞)(x) for x≥0 is concave. Furthermore, fis differentiable on [0,∞) with
nondecreasing and continuous derivative f0(x)=2x[0,1](x) + 2 (1,∞)(x) for x≥0.
For ε∈(0,1), by Markov’s inequality,
P
n
X
k=1
ckYk
> ε≤1
f(ε)Ef
n
X
k=1
ckYk≤4
ε2
n
X
k=1
E[f(|ck||Yk|)]
=4
ε2
n
X
k=1 |ck|Z∞
0
f0(|ck|x)P(|Y|> x) dx
≤4
ε2
n
X
k=1 |ck|Z∞
0
f0(cx)P(|Y|> x) dx
=8
ε2Z1/c
0
cxP(|Y|> x) dx+Z∞
1/c
P(|Y|> x) dx,
where we have used Lemma A.1 and f(ε) = ε2for ε∈(0,1) for the second in-
equality, and monotonicity of f0for the third. Integration by parts gives the first
equality.
A.2. The minimal position. In this section, we collect some known results con-
cerning the minimal position in a branching random walk in what is called the
boundary case, see [11].
Proposition A.3. Let ((V(u))|u|=n)n∈N0be a branching random walk such that
the positions in the first generation V(u),|u|= 1 satisfy the assumptions (2.11),
(2.14) and (2.15). Then the sequence of distributions of n3/2sup|u|=ne−V(u),n∈N
is tight. In particular,
n1/2sup
|u|=n
e−V(u)P
→0as n→ ∞.(A.6)
38 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
Notice that, under some extra non-lattice assumption, [1, Theorem 1.1] gives the
stronger statement
lim
n→∞ Pmin
|u|=nV(u)−3
2log n≥x=E[e−C∗exD∞] (A.7)
for all x∈Rwhere C∗is a positive constant and, as before, D∞is the limit of the
derivative martingale defined in (2.12). For our proposition, we do not need the
full strength of (A.7), which allows us to work without the lattice assumption.
Proof of Proposition A.3. We need to estimate min|v|=nVn(u) from below. Recall
that, for u∈ G, we write V(u):= mink=0,...,|u|V(u|k). With this notation, we have
Phmin
|v|=nVn(u)<−xi≤Pmin
u∈G V(u)≤ −x+Pmin
|u|=n,
V(u)≥−x
Vn(u)<−x
≤C(1 + x)e−x
by inequality (4.12) from [34] and [1, Corollary 3.4]. The latter does not require a
non-lattice assumption.
A.3. Asymptotic cancellation. Let Lu= (Lu(v))v∈N,u∈Vdenote a family
of i.i.d. copies of a sequence (L(v))v∈N= (L∅(v))v∈Nof complex-valued random
variables satisfying
#{v∈N:L(v)6= 0}<∞a. s. (A.8)
Define L(∅):= 1 and, recursively,
L(uv):=L(u)·Lu(v)
for u∈Vand v∈N. Further, we let
Zn:=X
|u|=n
L(u) and Wn:=X
|u|=n|L(u)|
where summation over |u|=nhere means summation over all u∈Nnwith L(u)6=
0. Finally, we set ˜
W1:=P|v|=1 |L(v)|log−(|L(v)|). We extend the shift-operator
notation introduced in Section 3.2 to the present context, so if X= Ψ((Lv)v∈V) is
a function of the whole family (Lv)v∈Vand u∈V, then [X]u:= Ψ((Luv)v∈V).
Lemma A.4. Assume that E[W1] = 1 and that a:=E[Z1]∈Csatisfies |a|<1.
(a) If E[P|v|=1 |L(v)|p]<1and E[Wp
1]<∞for some p > 1, then Zn→0a. s.
and in Lp∧2.
(b) Suppose that one of the following two conditions holds.
(i) Wn→Win L1;
(ii) E[P|v|=1 |L(v)|log(|L(v)|)] = 0,E[W1log2
+(W1)] <∞,E[˜
W1log+(˜
W1)] <∞,
EX
|v|=1 |L(v)|log2(|L(v)|)<∞.
Then Zn→0in probability if (i) holds and n1/2Zn→0in probability if (ii) holds.
Proof. (a) We can assume without loss of generality that p∈(1,2]. According to
[20, Corollary 5] or [31, Theorem 2.1] the martingale Wnconverges a.s. and in Lp
to some limit W. Let
q:= max{|a|p,E[P|v|=1 |L(v)|p]}<1.
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 39
For k=bn/2c, we have
E[|Zn|p]≤2p−1E[|Zn−akZn−k|p]+2p−1E[|akZn−k|p]
≤2p−1EEX
|v|=n−k
L(v)([Zk]v−ak)
pFn−k+ 2p−1|a|kp E[Wp
n−k]
≤2p+1EX
|v|=n−k|L(v)|pE[|Zk−ak|p]+ 2p−1qkE[Wp]
≤2p+1EX
|v|=n−k|L(v)|p2p−1(E[|Zk|p] + |a|kp)+2p−1qkE[Wp]
≤2p+1(E[Wp] + 1) + E[Wp]2p−1qk
where we have repeatedly used that |z+w|p≤2p−1(|z|p+|w|p) and Lemma A.1
for the third inequality. The bound decays exponentially as n→ ∞ giving Zn→0
in Lpand also Zn→0 a. s. by virtue of the Borel-Cantelli lemma and Markov’s
inequality.
(b) Let S:={Wn>0 for all n≥0}denote the survival set of the system. It is
clear that the claimed convergence holds on the set of extinction Sc. Therefore, in
what follows we work under P∗(·):=P(·|S).
We first assume that (i) holds. Then
Wn→WP∗-a. s. (A.9)
Eq. (A.9) in combination with (3.2) gives
sup|v|=n|L(v)|
Wn→0P∗-a. s. (A.10)
If, on the other hand, assumption (ii) is satisfied, then [2, Theorem 1.1] gives
√nWn→W∗in P∗-probability (A.11)
for some random variable W∗satisfying P∗(W∗>0) = 1. As before, we need
control over max|u|=n|L(u)|. A combination of (A.11) and Proposition A.3 gives
sup|v|=n|L(v)|
P|v|=n|L(v)|=n3/2sup|v|=n|L(v)|
n1/2Wn
1
n
P∗
→0 as n→ ∞.(A.12)
From now on, we treat both cases, (i) and (ii), simultaneously. In view of (A.9)
and (A.11), it remains to prove that
lim
n→∞
Zn
Wn
= 0 in P∗-probability. (A.13)
The last relation follows if we can show that, for any fixed positive integer k < n,
lim
n→∞
Zn−akZn−k
Wn
= 0 in P∗-probability (A.14)
and that, for all ε∈(0,1),
lim
k→∞ lim sup
n→∞
P∗
akZn−k
Wn
> ε= 0.(A.15)
40 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
Since, for any k∈N, we have
lim
n→∞
Wn−k
Wn
= 1 in P∗-probability,(A.16)
for all ksuch that |a|k< ε/2, we have
P∗
akZn−k
Wn
> ε≤P∗
akWn−k
Wn
> ε→0 as n→ ∞.
Hence, (A.15) holds and it remains to check (A.14). In view of (A.16), relation
(A.14) is equivalent to
lim
n→∞
Zn−akZn−k
Wn−k
= 0 in P∗-probability. (A.17)
Setting Sj:={Wj>0}for j∈N0, we have Sj↓ S as j→ ∞. Thus, for ε > 0, we
infer
P
Zn−akZn−k
Wn−k
> ε, S≤ESn−kP
Zn−akZn−k
Wn−k
> εFn−k,
so that it suffices to show that the right-hand side converges to zero. To this end,
we work on Sn−kwithout further notice. We use the representation
Zn−akZn−k
Wn−k
=X
|v|=n−k
L(v)
Wn−k
([Zk]v−ak).
Given Fn−k, the right-hand side is a weighted sum of i.i.d. centered complex-
valued random variables which satisfies the assumptions of Corollary A.2 with
cv=L(v)/Wn−k,|v|=n−kand Y=Zk−ak. Note that #{cv:cv6=
0,|v|=n−k}<∞a. s. in view of (A.8), that P|v|=n−k|cv|= 1 a. s. and that
E[|Zk−ak|]≤E[|Zk|] + |a|k≤E[Wk] + |a|k≤2. With
c(n−k):=sup|v|=n−k|L(v)|
P|v|=n−k|L(v)|
an application of Corollary A.2 yields
P
Zn−akZn−k
Wn−k
> εFn−k≤8
ε2c(n−k)Z1/c(n−k)
0
xP(|Zk−ak|> x) dx
+Z∞
1/c(n−k)
P(|Zk−ak|> x) dx.
We claim that the right-hand side converges to zero in probability as n→ ∞.
Indeed, according to (A.10) and (A.12), respectively, c(n−k)→0 in P∗-probability.
It remains to use the following simple fact. If his a measurable function satisfying
limy→0h(y) = 0 and if limn→∞ τn= 0 in probability, then limn→∞ h(τn) = 0 in
probability. For instance, apply this to h(y) = yR1/y
0xP(|Zk−ak|> x) dxand
τn=c(n−k). It follows from Markov’s inequality and E[|Zk−ak|]<∞that
limx→∞ xP(|Zk−ak|> x) = 0, which in turn implies h(y)→0 as y→0. The
proof of (A.17), and hence of (A.14), is complete.
FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS 41
Acknowledgements. The research of K. K. and M. M. was supported by DFG Grant
ME 3625/3-1. K. K. was further supported by the National Science Center, Poland
(Sonata Bis, grant number DEC-2014/14/E/ST1/00588). A part of this work was
done while A. I. was visiting Innsbruck in August 2017. He gratefully acknowledges
hospitality and the financial support again by DFG Grant ME 3625/3-1.
References
[1] Elie A¨ıd´ekon. Convergence in law of the minimum of a branching random walk. Ann. Probab.,
41(3A):1362–1426, 2013.
[2] Elie A¨ıd´ekon and Zhan Shi. The Seneta-Heyde scaling for the branching random walk. Ann.
Probab., 42(3):959–993, 2014.
[3] G. Alsmeyer, A. Iksanov, S. Polotskiy, and U. R¨osler. Exponential rate of Lp-convergence
of intrinsic martingales in supercritical branching random walks. Theory Stoch. Process.,
15(2):1–18, 2009.
[4] Gerold Alsmeyer, J. D. Biggins, and Matthias Meiners. The functional equation of the
smoothing transform. Ann. Probab., 40(5):2069–2105, 2012.
[5] Gerold Alsmeyer and Alexander Iksanov. A log-type moment result for perpetuities and its
application to martingales in supercritical branching random walks. Electron. J. Probab.,
14:no. 10, 289–312, 2009.
[6] J. D. Biggins. Martingale convergence in the branching random walk. J. Appl. Probability,
14(1):25–37, 1977.
[7] J. D. Biggins. Uniform convergence of martingales in the branching random walk. Ann.
Probab., 20(1):137–151, 1992.
[8] J. D. Biggins and A. E. Kyprianou. Seneta-Heyde norming in the branching random walk.
Ann. Probab., 25(1):337–360, 1997.
[9] John D. Biggins. Lindley-type equations in the branching random walk. Stochastic Process.
Appl., 75(1):105–133, 1998.
[10] John D. Biggins and Andreas E. Kyprianou. Measure change in multitype branching. Ad-
vances in Applied Probability, 36(2):544–581, 2004.
[11] John D. Biggins and Andreas E. Kyprianou. Fixed points of the smoothing transform: the
boundary case. Electron. J. Probab., 10:no. 17, 609–631 (electronic), 2005.
[12] Patrick Billingsley. Convergence of probability measures. John Wiley & Sons, Inc., New York-
London-Sydney, 1968.
[13] Wolfgang J. B¨uhler. Ein zentraler Grenzwertsatz f¨ur Verzweigungsprozesse. Z. Wahrschein-
lichkeitstheorie und Verw. Gebiete, 11:139–141, 1969.
[14] Dariusz Buraczewski, Ewa Damek, Yves Guivarc’h, Andrzej Hulanicki, and Roman Urban.
Tail-homogeneity of stationary measures for some multidimensional stochastic recursions.
Probab. Theory Related Fields, 145(3-4):385–420, 2009.
[15] Dariusz Buraczewski, Ewa Damek, Sebastian Mentemeier, and Mariusz Mirek. Heavy tailed
solutions of multivariate smoothing transforms. Stochastic Process. Appl., 123(6):1947–1986,
2013.
[16] Lisa Hartung and Anton Klimovsky. The phase diagram of the complex branching Brownian
motion energy model, 2017.
[17] Inge S. Helland. Central limit theorems for martingales with discrete or continuous time.
Scand. J. Statist., 9(2):79–94, 1982.
[18] C. C. Heyde. A rate of convergence result for the super-critical Galton-Watson process. J.
Appl. Probability, 7:451–454, 1970.
[19] C. C. Heyde and B. M. Brown. An invariance principle and some convergence rate results for
branching processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 20:271–278, 1971.
[20] Alexander Iksanov. Elementary fixed points of the BRW smoothing transforms with infinite
number of summands. Stochastic Process. Appl., 114(1):27–50, 2004.
[21] Alexander Iksanov and Zakhar Kabluchko. A central limit theorem and a law of the iterated
logarithm for the Biggins martingale of the supercritical branching random walk. J. Appl.
Probab., 53(4):1178–1192, 2016.
[22] Alexander Iksanov, Konrad Kolesko, and Matthias Meiners. Stable-like fluctuations of Big-
gins’ martingales, 2017. arXiv:1709.07362.
42 FLUCTUATIONS OF BIGGINS’ MARTINGALES AT COMPLEX PARAMETERS
[23] Alexander Iksanov, Xingang Liang, and Quansheng Liu. On Lp-convergence of the Biggins
martingale with complex parameter, 2018. Work in progress.
[24] Alexander Iksanov and Matthias Meiners. Exponential rate of almost-sure convergence of
intrinsic martingales in supercritical branching random walks. J. Appl. Probab., 47(2):513–
525, 2010.
[25] O. M. ¯
Iksanov. On the rate of convergence of a regular martingale related to a branching
random walk. Ukra¨ın. Mat. Zh., 58(3):326–342, 2006.
[26] Peter Jagers. General branching processes as Markov fields. Stochastic Process. Appl.,
32(2):183–212, 1989.
[27] Olav Kallenberg. Foundations of modern probability. Probability and its Applications (New
York). Springer-Verlag, New York, second edition, 2002.
[28] Konrad Kolesko. Tail homogeneity of invariant measures of multidimensional stochastic re-
cursions in a critical case. Probab. Theory Related Fields, 156(3-4):593–612, 2013.
[29] Konrad Kolesko and Matthias Meiners. Convergence of complex martingales in the branching
random walk: the boundary. Electron. Comm. Probab., 18(22):1–14, 2017.
[30] Thomas G. Kurtz. Inequalities for the law of large numbers. Ann. Math. Statist., 43:1874–
1883, 1972.
[31] Quansheng Liu. On generalized multiplicative cascades. Stochastic Process. Appl., 86(2):263–
286, 2000.
[32] Russell Lyons. A simple path to Biggins’ martingale convergence for branching random walk.
In Classical and modern branching processes (Minneapolis, MN, 1994), volume 84 of IMA
Vol. Math. Appl., pages 217–221. Springer, New York, 1997.
[33] Thomas Madaule. The tail distribution of the derivative martingale and the global minimum
of the branching random walk. pages 1–25, 2016. arXiv:1606.03211.
[34] Thomas Madaule. Convergence in law for the branching random walk seen from its tip. J.
Theoret. Probab., 30(1):27–63, 2017.
[35] Pascal Maillard and Michel Pain. 1-stable fluctuations in branching Brownian motion at
critical temperature I: the derivative martingale, 2018. arXiv:1806.05152.
[36] Matthias Meiners and Sebastian Mentemeier. Solutions to complex smoothing equations.
Probab. Theory Related Fields, 168(1-2):199–268, 2017.
[37] Zhan Shi. Branching random walks, volume 2151 of Lecture Notes in Mathematics. Springer,
Cham, 2015. Lecture notes from the 42nd Probability Summer School held in Saint Flour,
2012, ´
Ecole d’´
Et´e de Probabilit´es de Saint-Flour. [Saint-Flour Probability Summer School].
[38] Frank Spitzer. Principles of random walk. Springer-Verlag, New York-Heidelberg, second
edition, 1976. Graduate Texts in Mathematics, Vol. 34.
[39] V. A. Vatutin and V. A. Topchi˘ı. The maximum of critical Galton-Watson processes, and
left-continuous random walks. Teor. Veroyatnost. i Primenen., 42(1):21–34, 1997.
[40] D.V. Widder. The Laplace transform. Princeton mathematical series. Princeton university
press, 1946.
