Simulation of Weakly Self-Similar Stationary Increment {\text{Sub}}_{\varphi } {\left( \Omega \right)}-Processes: A Series Expansion Approach

Abstract

We consider simulation of \({\text{Sub}}_{\varphi } {\left( \Omega \right)}\)-processes that are weakly selfsimilar with stationary increments in the sense that they have the covariance function
$$R{\left( {t,s} \right)} = \frac{1}{2}{\left( {t^{{2H}} + s^{{2H}} - {\left| {t - s} \right|}^{{2H}} } \right)}$$for some H ∈ (0, 1). This means that the second order structure of the processes is that of the fractional Brownian motion. Also, if \(H >\frac{1} {2}\) then the process is long-range dependent.
The simulation is based on a series expansion of the fractional Brownian motion due to Dzhaparidze and van Zanten. We prove an estimate of the accuracy of the simulation in the space C([0, 1]) of continuous functions equipped with the usual sup-norm. The result holds also for the fractional Brownian motion which may be considered as a special case of a \({\text{Sub}}_{{{x^{2} } \mathord{\left/ {\vphantom {{x^{2} } 2}} \right. \kern-\nulldelimiterspace} 2}} {\left( \Omega \right)}\)-process.

Full text

Simulation of Weakly Self-Similar Stationary
Increment Sub’’’’’’ððððÞÞÞÞÞð -Processes: A Series
Expansion Approach
YURIY KOZACHENKO yvk@univ.kiev.ua
Mechanics and Mathematics Faculty, Department of Probability Theory and Math. Statistics, Taras Shevchenko
Kyiv National University, Volodymyrska 64, Kyiv, Ukraine
TOMMI SOTTINEN tommi.sottinen@helsinki.fi
Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Ha¨llstro¨min katu 2b),
00014 Helsinki, Finland
OLGA VASYLYK vasylyk@univ.kiev.ua
Mechanics and Mathematics Faculty, Department of Probability Theory and Math. Statistics, Taras Shevchenko
Kyiv National University, Volodymyrska 64, Kyiv, Ukraine
Received November 10, 2004; Revised October 5, 2005; Accepted July 6, 2005
Abstract. We consider simulation of Sub’ðÞ-processes that are weakly selfsimilar with stationary increments
in the sense that they have the covariance function
Rt;sðÞ¼
1
2t2Hþs2Hts
jj
2H

or some H2(0, 1). This means that the second order structure of the processes is that of the fractional
Brownian motion. Also, if H>1
2then the process is long-range dependent.
The simulation is based on a series expansion of the fractional Brownian motion due to Dzhaparidze and van
Zanten. We prove an estimate of the accuracy of the simulation in the space C([0, 1]) of continuous functions
equipped with the usual sup-norm. The result holds also for the fractional Brownian motion which may be
considered as a special case of a Subx2=2ðÞ-process.
Keywords: fractional Brownian motion, ’-sub-Gaussian processes, long-range dependence, self-similarity,
series expansions, simulation
AMS 2000 Subject Classification: 60G18, 60G15, 68U20, 33C10
1. Introduction
We consider simulation of centred second order processes defined on the interval [0, 1]
whose covariance function is
Rt;sðÞ¼
1
2t2Hþs2Hts
jj
2H

;
and belong to the space Sub’ðÞ. This space is defined later in Section 2. The parameter
Htakes values in the interval (0,1) the other cases being either uninteresting or impossible.
Methodology and Computing in Applied Probability, 7, 379–400, 2005
#2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands.

The motivation to study processes with the second order structure given by Rcomes
from the notions of statistical self-similarity and long-range dependence. A square
integrable process with stationary increments is long-range dependent if its autocor-
relation function is not summable. A process Zis self-similar with index Hif it satisfies
the scaling property
Zt
ðÞ
t0¼
daHZat

t0
for all a> 0. Here dmeans equality in distributions. The self-similarity parameter H2
(0, 1), or Hurst index, has also the following role. If H6¼ 1
2then Zis a process with
dependent increments. There are 1
2-self-similar processes with independent increments,
but these are not square integrable processes. If H>1
2then the increments of the process
Zare long-range dependent. The case H<1
2corresponds to short-range dependence.
These properties, self-similarity and long-range dependence, have been shown to be
charateristic in e.g., teletraffic and financial time series. See (Beran, 1994; Doukhan
et al., 2003; Embrechts and Maejima, 2002; Samorodnitsky and Taqqu, 1994) for refer-
ences to these studies and for self-similarity and long-range dependence in general.
Assume now that a process Zis H-self-similar, has stationary increments, and is
centred and square integrable. Then it is easy to see that Zhas Ras the covariance
function. So, if a process has the covariance function Rwe say that it is weakly self-
similar with stationary increments, or second order self-similar with stationary in-
crements. In the Gaussian case the properties of the weak self-similarity and the proper
one coincide. In this case Zis called the fractional Brownian motion, and, in particu-
lar, the Brownian motion if H¼1
2
. The fractional Brownian motion was originally
defined and studied by Kolmogorov (1940) under the name BWiener helix.^The name
Bfractional Brownian motion^comes from Mandelbrot and Van Ness (1968).
Recently Dzhaparidze and van Zanten (2004) proved a series representation for the
fractional Brownian motion B:
Bt¼X
1
n¼1
sin xntðÞ
xn
XnþX
1
n¼1
1cos yntðÞ
yn
Yn:ð1:1Þ
Here the X
n
’s and the Y
n
’s are independent zero mean Gaussian random variables with
certain variances depending on Hand n.Thex
n
’s are the positive real zeros of the Bessel
function J
jH
of the first kind and the y
n
’s are the positive real zeros of the Bessel
function J
1jH
. The series in (1.1) converge in mean square as well as uniformly on [0, 1]
with probability 1. Details of representation (1.1) are recalled later in Section 3.
In this paper we study the use of the expansion (1.1) in simulating processes with the
covariance function R. In particular, we study processes of the form (1.1) where the X
n
’s
and Y
n
’s are replaced by independent random variables from the space Sub’ðÞ. The
fractional Brownian motion may be obtained as a special case with ’(x)=x
2
/2.
Let us end this introduction by saying a few words of the pros and cons of using the
series expansion (1.1). The Hurst parameter His roughly the Ho¨lder index of the process.
This means that, especially in the case of small H, the sample paths of the process are
very erratic. However, the coefficient functions in (1.1) are smooth. So, in order to have
380 KOZACHENKO, SOTTINEN AND VASYLYK

good accuracy in simulation one needs a large truncation point in the expansion. This is
the bad news. The good news is that once the coefficient functions are calculated we are
in no way restricted to any pregiven time grid. Indeed, unlike in some traditional simu-
lation methods, to calculate the value of the sample path in a new time point one does
not have to condition on the already calculated time points. The computational effort in
adding a new time point is always constant.
2. Space Sub’’’’’’’ððððÞÞÞÞ
We recall some basic facts about the space Sub’ðÞof ’-sub-Gaussian (or generalised
sub-Gaussian) random variables. For details and proofs we refer to Buldygin and
Kozachenko (2000) and Krasnoselskii and Rutitskii (1958).
DEFINITION 2.1 (Krasnoselskii and Rutitskii, 1958) A continuous even convex function
u¼uxðÞ;x2R
fg
is an Orlicz N-function if it is strictly increasing for x > 0, u(0) = 0
uxðÞ
x!0asx!0 and uxðÞ
x!1 as x!1:
PROPOSITION 2.2 (Krasnoselskii and Rutitskii, 1958) The function u is an Orlicz N-
function if and only if
uxðÞ¼Zx
jj
0
luðÞdu;x2R;
where the density function l is nondecreasing, right continuous, l(u)>0as u >0,l(0) = 0
and l(u)YVas u YV.
DEFINITION 2.3 Let ube an Orlicz N-function. The even function u*¼u*xðÞ;x2R
fg
defined by the formula
u*xðÞ¼sup
y>0
xy uyðÞðÞ;x0;
is the Young-Fenchel transformation of the function u.
PROPOSITION 2.4 (Krasnoselskii and Rutitskii, 1958) The function u* is an Orlicz N-
function and for x >0
u*xðÞ¼xy0uy
0
ðÞif y0¼l1xðÞ:
Here l
j1
is the generalised inverse function of l, i.e.,
l1xðÞ:¼sup v0:lvðÞx
fg
:
SIMULATION OF WEAKLY SELF-SIMILAR STATIONARY INCREMENT SUB’ðÞ-PROCESSES 381

DEFINITION 2.5 The assumption Q holds for an Orlicz N-function ’if it is quadratic
around the origin, i.e., there exist such constants x
0
>0and C > 0that ’(x) = Cx
2
for
ªxªex
0
.
EXAMPLE 2.6 The assumption Qholds for the following Orlicz N-functions
’xðÞ¼
x
jj
p
pif xjj>1;
x2
pif x
jj1;
(p>1;
’xðÞ¼
e
2

2
x2if x
jj2


1=;
exp x
jj

fg
if x
jj>2


1=;
(0<<1:
Let (,F,P) be a standard probability space.
DEFINITION 2.7 Let ’be an Orlicz N-function satisfying the assumption Q. A zero mean
random variable belongs to the space Sub’ðÞ,the space of ’-sub-Gaussian random
variables, if there exists a positive and finite constant asuch that the inequality
Eexp 
fg
exp ’aðÞ
fg
holds for all 2R.
REMARK 2.8 Note that like the Gaussian variables the ’-sub-Gaussian random variables
also have light tails. In particular, they have moments of all orders.
PROPOSITION 2.9 (Buldygin and Kozachenko, 2000) The space Sub’ðÞis a Banach space
with respect to the norm
’ðÞ¼inf aQ0:Eexp 
fg
exp ’aðÞ
fg
;2R
fg
:
Moreover, for any 2Rwe have
Eexp 
fg
exp ’
’ðÞ

;E2

1
22CðÞ
1
2’ðÞ;
where C is the constant from the assumption Q.
DEFINITION 2.10 A stochastic process X =(X
t
)
t2[0,1]
is a Sub’ðÞ-process if it is a
bounded family of Sub’ðÞrandom variables: Xt2Sub’ðÞfor all t 2[0, 1] and
sup
t20;1½
’Xt
ðÞ<1:
The properties of random variables from the spaces Sub’ðÞwere studied in the book
(Buldygin and Kozachenko, 2000).
REMARK 2.11 When ’xðÞ¼x2
2the space Sub’ðÞis called the space of sub-Gaussian
random variables and is denoted by Sub(). Centred Gaussian random variables belong
to the space Sub(), and in this case ’ðÞis just the standard deviation: (E
2
)
1/2
. Also, if
is bounded, i.e., ªªeca.s. then 2Sub() and 
’
()ec.
382 KOZACHENKO, SOTTINEN AND VASYLYK

PROPOSITION 2.12 Let ’be an Orlicz N-function satisfying the assumption Q. Assume
further that the function ’ffiffiffi

p

is convex. Let 
1
,
2
,..., 
n
be independent random
variables from the space Sub’ðÞ.Then
2
’X
n
i¼1
i
!
X
n
i¼1
2
’i
ðÞ:
3. Series Representation
Let us now recall the DzhaparidzeYvan Zanten series representation (1.1) in detail. Let J
be the Bessel function of the first kind of order , i.e.,
JxðÞ¼X
1
n¼0
1ðÞ
nx=2ðÞ
þ2n
nþ1ðÞþnþ1ðÞ
:
Here x>0,mj1, j2, . . . and Gdenotes the Euler Gamma function
zðÞ¼Z1
0
tz1etdt:
It is well-known that for >j1 the Bessel function Jhas countable number of real
positive zeros tending to infinity. Denote by x
n
the nth positive real zero of the Bessel
function J
jH
;y
n
is the nth positive real zero of J
1jH
.
Let Bbe the fractional Brownian motion with index H. Then it may be represented as
the mean square convergent series
Bt¼X
1
n¼1
cnsin xntðÞ
e
XXnþX
1
n¼1
dn1cos yntðÞðÞ
e
YYn:
Here e
XXn;e
YYn;n¼1;2;:::; are independent and identically distributed zero mean
Gaussian random variables with Ee
XX 2
n¼Ee
YY 2
n¼1 and
cn¼Hffiffiffiffiffi
2c
p
xHþ1
nJ1Hxn
ðÞ
;n¼1;2;:::; ð3:1Þ
dn¼Hffiffiffiffiffi
2c
p
yHþ1
nJHyn
ðÞ
;n¼1;2;:::; ð3:2Þ
c¼2Hþ1ðÞsin HðÞ
2Hþ1:ð3:3Þ
We shall generalise the setting above in the following way: Define a process
Z=(Zt)
t2[0,1]
by the expansion
Zt¼X
1
n¼1
cnsin xntðÞnþX
1
n¼1
dn1cos yntðÞðÞn;ð3:4Þ
SIMULATION OF WEAKLY SELF-SIMILAR STATIONARY INCREMENT SUB’ðÞ-PROCESSES 383

where c
n
and d
n
are given by (3.1) and (3.2), x
n
,h
n
,n=1,2,..., are independent
identically distributed centred random variables from the space Sub’ðÞwith E
n
2
=
Eh
n
2
=1,n=1, 2, .... Furthermore, we shall assume that the function ’ffiffi
p
ðÞis convex.
Since ’-sub-Gaussian random variables are square integrable we have the following
direct consequence of the series representation (1.1) for fractional Brownian motion.
PROPOSITION 3.1 The series (3.4) converges in mean square and the covariance function
of the process Z is R.
In addition to the L
2
-convergence the spaces Sub’ðÞare nice enough to allow
uniform w-by-wconvergence.
THEOREM 3.2 The series (3.4) converges uniformly with probability one and the process
Z is almost surely continuous on [0, 1]. Moreover, if Z is strongly self-similar with
stationary increments then it is -Ho¨lder continuous with any index <H.
The continuity in Theorem 3.2 follows by using Hunt’s theorem (Hunt, 1951). The
Ho¨lder continuity comes from Kolmogorov’s criterion (Embrechts and Maejima, 2002),
Lemma 4.1.1. Let us also note that from the case of fractional Brownian motion we know
that in general we cannot have Ho¨lder continuity with index =H, cf. (Decreusefond
and U
¨stu¨nel, 1999).
For the reader’s convenience we now recite a modification of Hunt’s theorem as a
lemma (cf. Buldygin and Kozachenko, 2000, Example 3.5.2).
LEMMA 3.3 Suppose that (
n
)
nQ1
is a sequence of independent centred random variables
with E
n
2
=1,n=1,2,....Let (
n
)
nQ1
be a sequence such that 
n
e
n+1
and 
n
YV
as n YV.
If
X
1
n¼1
a2
nln 1 þn
ðÞðÞ
1þ<1
for some >0then the series
X
1
n¼1
ancos ntðÞnand X
1
n¼1
ansin ntðÞn
converge uniformly on [0,1] with probability one.
Proof of Theorem 3.2: Let us consider the almost sure uniform convergence first. Now,
from Watson (1944), p. 506, we have x
n
õy
n
õnas nYV. Also from (Watson, 1944),
p. 200, we have the following asymptotic relation for the Bessel function Jfor >j1:
J2
xðÞþJ2
þ1xðÞõ2
x
384 KOZACHENKO, SOTTINEN AND VASYLYK

for large ªxª. Since the zeros x
n
of J
jH
and y
n
of J
1jH
tend to infinity this yields
J2
1Hxn
ðÞõ2
xn
and J2
Hyn
ðÞõ2
yn
as nYV. Therefore,
c2
nõc
n2Hþ1and d2
nõc
n2Hþ1
(see (3.1)Y(3.3)). Consequently, the series
X
1
n¼1
c2
nln 1 þxn
ðÞðÞ
1þ"and X
1
n¼1
d2
nln 1 þyn
ðÞðÞ
1þ"
converge for all "> 0. The almost sure uniform convergence and the continuity of the
process follow now from Hunt’s theorem (Lemma 3.3), since almost sure uniform
convergence of the series P1
n¼1dncos yntðÞnand almost sure convergence of the series
P1
n¼1dnnimply that P1
n¼1dn1cos yntðÞðÞnalso converges uniformly on [0, 1] with
probability one.
To see the Ho¨lder continuity of Zjust use strong self-similarity and the stationarity of
the increments together with the fact that Zhas all moments. Indeed, for all n2Nwe
have
EZtZs
jj
n¼EZts
jj
n¼ts
jj
HnEZ1
jj
n;
and the claim follows from Kolmogorov’s criterion. Í
4. Simulation, Accuracy and Reliability
We want to construct a model e
ZZ of the process Z, such that e
ZZ approximates Zwith given
reliability and accuracy in the norm of some Banach space. In this paper we consider the
space C([0, 1]) equipped with the usual sup-norm.
DEFINITION 4.1 The model e
ZZ approximates the process Z with given reliability 1 j,0<
<1,and accuracy >0inC([0, 1]) if
Psup
t20;1½
Zte
ZZt
>
!
:
A natural model for Z, defined by the expansion (3.4), would be the truncated series
X
N
n¼1
cnsin xntðÞnþdn1cos yntðÞðÞn
ðÞ:
However, it is realistic to assume that the constants c
n
and d
n
and the zeros x
n
,y
n
are only
calculated approximately, especially since there are fast-to-compute asymptotic formulas
for the zeros x
n
and y
n
(cf. Watson, 1944, p. 506). Note that the constants c
n
and d
n
depend on the zeros.
SIMULATION OF WEAKLY SELF-SIMILAR STATIONARY INCREMENT SUB’ðÞ-PROCESSES 385

Let e
ccnand e
ddnbe the approximated values of the c
n
and d
n
, respectively. Let
e
ccncn
jj
c
n;
je
ddndnjd
n;
n= 1, ... , N. The errors 
n
c
and 
n
d
are assumed to be known. Let e
xxnand e
yynbe
approximations of the corresponding zeros x
n
and y
n
with error bounds
e
xxnxn
jj
x
n;
e
yynyn
jj
y
n;
The error bounds 
n
x
and 
n
y
are also assumed to be known.
Then, the model of the process Zis
e
ZZt¼X
N
n¼1e
ccnsin e
xxntðÞnþe
ddn1cos e
yyntðÞðÞn

:ð4:1Þ
The error in the simulation (model) is
t:¼Zte
Zt
Zt
¼X
N
n¼1
cnsin xntðÞ
e
ccnsin e
xn
xntðÞðÞnþdn1cos yntðÞðÞ
e
ddn1cos e
yyntðÞðÞ

n
no
þX
1
n¼Nþ1
cnsin xntðÞnþdn1cos yntðÞðÞn
fg
¼:
appr
tþcut
t:
In order to bound the error Din C([0, 1]) we need estimates for ’t
ðÞand
’ts
ðÞfor all s,t2[0, 1]. The estimates are given in the following proposition.
PROPOSITION 4.2 Denote a’:¼’n
ðÞ¼’n
ðÞand
cut :¼a2
’X
1
n¼Nþ1
c2
nþ4d2
n

;
appr :¼a2
’X
N
n¼1
cnx
nþc
n

2þdny
nþ2d
n

2
no
:
Let 2(0, H)and denote
cut
:¼222a2
’X
1
n¼Nþ1
c2
nx2
nþd2
ny2
n

;
appr
:¼232a2
’X
N
n¼1
x2
nc
n

2þy2
nd
n

2þ232e
cn
cn
ðÞ
2x
n

2xnþ~
xxn
ðÞ
2
22þ1
(
þe
ddn

2y
n

2ynþ~
yyn

2
22þ1!):
386 KOZACHENKO, SOTTINEN AND VASYLYK

Then we have for all s, t 2[0, 1]
2
’t
ðÞappr þcut ;ð4:2Þ
2
’ts
ðÞappr
þcut


ts
jj
2:ð4:3Þ
Proof: By Proposition 2.12 we know that
2
’t
ðÞ2
’appr
t
ðÞþ2
’cut
t

:ð4:3Þ
For 2
’appr
t
ðÞwe obtain by using Proposition 2.12 and the mean value theorem that
2
’appr
t
ðÞ
X
N
n¼1
cnsin xntðÞ
e
ccnsin e
xxntðÞðÞ
22
’n
ðÞ
þX
N
n¼1
dn1cos yntðÞðÞ
e
ddn1cos e
yyntðÞðÞ

22
’n
ðÞ
¼a2
’X
N
n¼1
cnsin xntðÞsin e
xxntðÞðÞþcne
ccn
ðÞsin e
xxntðÞðÞ
2
(
þX
N
n¼1
dncos e
yyntðÞcos yntðÞðÞþdne
ddn

1cos e
yyntðÞðÞ

2
a2
’X
N
n¼1
cnxne
xxn
jj
tþcne
ccn
ðÞsin e
xxntðÞ
jj
ðÞ
2
(
þX
N
n¼1
dne
yynyn
jjtþdne
ddn

1cos e
yyntðÞðÞ


2
a2
’X
N
n¼1
cnx
nþc
n

2þdny
nþ2d
n

2
no
:
¼appr:
Similarly we obtain
2
’cut
t

a2
’X
1
n¼Nþ1
c2
nþ4d2
n

¼cut:ð4:4Þ
Recall the asymptotics of c
n
2
and d
n
2
(3.5) to see that the series (4.4) above converges.
The estimate (4.2) follows.
SIMULATION OF WEAKLY SELF-SIMILAR STATIONARY INCREMENT SUB’ðÞ-PROCESSES 387

Now we shall estimate the incremental error 2
’ts
ðÞ. For the Bcut-off^part
we have
2
’cut
tcut
s

¼2
’X
1
n¼Nþ1
cnsin xntðÞsin xnsðÞðÞnþX
1
n¼Nþ1
dncos ynsðÞcos yntðÞðÞn
!
221ðÞ
a2
’X
1
n¼Nþ1
c2
nxnts
jj
ðÞ
2þd2
nynts
jj
ðÞ
2

¼221ðÞ
a2
’X
1
n¼Nþ1
c2
nx2
nþd2
ny2
n

ts
jj
2
¼cut
ts
jj
2:ð4:5Þ
Due to the asymptotics (3.5) and x
n
õy
n
õnthe series in (4.5) converge if <H.
For the Bapproximating part^we have
2
’appr
tappr
s

a2
’X
N
n¼1
cnsin xntðÞsin xnsðÞðÞ
e
ccnsin e
xxntðÞsin e
xxnsðÞðÞðÞ
2
(
þX
N
n¼1
dncos ynsðÞcos yntðÞðÞ
e
ddncos e
yynsðÞcos e
yyntðÞðÞ

2):ð4:6Þ
By using the relations
ab cdjjeajjbdjjþacjjdjj;
aþbðÞ
2e2a2þ2b2;
sin xsin y¼2cos xþy
2sin xy
2
cos xcos y¼2sin xþy
2sin yx
2
we obtain for the summand in (4.6) that
cnsin xntðÞsin xnsðÞðÞ
e
ccnsin e
xxntðÞsin e
xxnsðÞðÞ
jj
sin xntðÞsin xnsðÞ
jj
cne
ccn
jj
þsin xntðÞsin xnsðÞsin e
xxntðÞþsin e
xxnsðÞ
jj
e
ccn
jj
;
sin xntðÞsin xnsðÞ
jj
¼2sin xntsðÞ
2

cos xntþsðÞ
2


2xntsðÞ
2


388 KOZACHENKO, SOTTINEN AND VASYLYK

and
sin xntðÞsin xnsðÞsin e
xxntðÞþsin e
xxnsðÞ
jj
¼sin xntðÞsin e
xxntðÞðÞsin xnsðÞsin e
xxnsðÞðÞ
jj
¼2sin xne
xxn
ðÞt
2

cos xnþe
xxn
ðÞt
2

:
2sin xne
xxn
ðÞs
2

cos xnþe
xxn
ðÞs
2


2 sin xne
xxn
ðÞt
2

cos xnþe
xxn
ðÞt
2

cos ðxne
xxnÞs
2!
þsin xne
xxn
ðÞt
2sin xne
xxn
ðÞs
2

cos xnþe
xxn
ðÞs
2



¼2sin xne
xxn
ðÞt
2

2sin xnþe
xxn
ðÞtþsðÞ
4

sin xnþe
xxn
ðÞstðÞ
4

½
þ2cos xnþe
xxn
ðÞtþsðÞ
4

sin xne
xxn
ðÞðtsÞ
4cos xnþe
xxn
ðÞs
2

4
xne
xxn
ðÞt
2
xnþe
xxn
ðÞðstÞ
4


þ
xne
xxn
ðÞtsðÞ
4

222x
n

xnþe
xxn
ðÞ

2þ1

ts
jj
:
Hence
cnsin xntsin xnse
ccnsin e
xxntsin e
xxns2
2xntsðÞ
2
c
nþ222x
n

xnþ~
xxn
ðÞ

2þ1

ts
jj
e
ccn
jj

2
22
xntsðÞ
2
c
n

2þ22
22x
n

xnþ~
xxn
ðÞ

2þ1

ts
jj
e
ccn
jj

2
¼ts
jj
2232x2
nc
n

2þ254e
ccn
ðÞ
2x
n

2xnþ~
xxn
ðÞ

2þ1

2

ts
jj
2232x2
nc
n

2þ254e
ccn
ðÞ
2x
n

22xnþ~
xxn
ðÞ

2

2þ2

¼tsjj
2232x2
nc
n

2þ264e
ccn
ðÞ
2x
n

2xnþ~xxn
ðÞ
2
22þ1

¼232ts
jj
2x2a
nc
n

2þ232e
ccn
ðÞ
2x
n

2xnþ~xxn
ðÞ
2
22þ1

:

SIMULATION OF WEAKLY SELF-SIMILAR STATIONARY INCREMENT SUB’ðÞ-PROCESSES 389

In the same way
dncos ynsðÞcos yntðÞðÞ
e
dncos e
yynsðÞcos e
yyntðÞðÞ

2
cos ynsðÞcos yntðÞ
jj
dne
dn
þcos ynsðÞcos yntðÞcos e
yynsðÞþcos e
yyntðÞ
jj
e
dn
2
21yn
jj
ts
jj
d
nþ222y
n

ts
jj
ynþ~yn
ðÞ

2þ1

e
ddn


2
22
1yn
jj
ts
jj
d
n

2þ22
22y
n

ts
jj
ynþ~yn
ðÞ

2þ1

e
dn


2
ts
jj
2232y2
nd
n

2þ254e
dn

2y
n

22ynþ~yn
ðÞ
2
22þ2


¼232ts
jj
2y2
nd
n

þ232e
dn

2y
n

2ynþ~yn
ðÞ
2
22þ1


:
Thus, we obtain following estimate
2
’appr
tappr
s

a2
’P
N
n¼1
232
ts
jj
2x2
nc
n

2þ232e
ccn
ðÞ
2x
n

2xnþ~xn
ðÞ
2
22þ1

þP
N
n¼1
232ts
jj
2y2
nd
n

2þ232e
ddn

2y
n

2ynþ~yn
ðÞ
2
22þ1


¼232a2
’ts
jj
2P
N
n¼1
x2
n
c
n

2þy2
nd
n

2
þ232~
ccn
ðÞ
2x
n

2xnþ~xn
ðÞ
2
22þ1

þe
ddn

2y
n

2ynþ~yn
ðÞ
2
22þ1


¼appr
tsjj
2:
ð4:7Þ
Estimate (4.3) follows now by collecting the estimates above and by using Proposition
2.12. Í
Now we are ready to state, although not yet to prove, our main result.
THEOREM 4.3 Let b and be such that 0<b<<H.Let 
appr
,

appr
,
cut
and 

cut
be as
in Proposition 4.2. Denote
0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
appr þcut
p;
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
appr
þcut

p;
¼min 0;
2
no
:ð4:7Þ
Let l be the density of ’.
The model e
ZZ, defined by (4.1), approximates the process Z, defined by ( 3.4), with given
reliability 1 j,0<< 1, and accuracy > 0 in C([0, 1]) if the following three
inequalities are satisfied:
0<; ð4:8Þ
390 KOZACHENKO, SOTTINEN AND VASYLYK

0

<
2exp ’1ðÞ
fg
1ðÞ
;ð4:9Þ
2 exp ’*
01

1
2b1b



0

b

l1
01

þ1
!
2
b
: ð4:10Þ
The following lemma is our main tool for proving Theorem 4.3. For the proof of it we
refer to Kozachenko and Vasilik (1998), Lemma 3.3.
LEMMA 4.4 Let X =(Xt)
t2[0, 1]
be a separable random process from the space Sub’ðÞ.
Let :R
+
YR
+
be a strictly increasing continuous function such that (h)Y0as h Y
0and
sup
ts
jj
h
’XtXs
ðÞhðÞ:
Denote 
0
= sup
t2[0,1]
’Xt
ðÞand let be such a number that 1
2
. Let
r:1;1½Þ!Rþbe a nondecreasing continuous function such that r(1) = 0 and the
mapping u 7!re
u
ðÞis convex. Suppose that
Z
0
uðÞdu <1;
where
uðÞ¼’;;r;uðÞ¼
rN1uðÞðÞðÞ
’1ln N1uðÞðÞðÞ
;
and N(")is the minimum number of closed intervals of the radius "that is needed to
cover the interval [0,1] (note that N "ðÞ1
2"þ1).
Then for all 2Rand p 2(0,1) we have
Eexp sup
t20;1½
Xt
jj
()
2exp ’0
1p

1pðÞþ’
1p

p

r10pðÞþ 
1pðÞpZp2
0
uðÞdu
0
B
@1
C
A
2
:ð4:11Þ
REMARK 4.5 (Buldygin and Kozachenko, 2000) Random process Xis called a separable
on (T, r)orr-separable, if there exist countable everywhere dense (with respect to
SIMULATION OF WEAKLY SELF-SIMILAR STATIONARY INCREMENT SUB’ðÞ-PROCESSES 391

pseudometric (metric) r) set STand a set 
0
Î:P(
0
) = 0, such that for any open
set UÎTand for any closed set DÎRwe have
\
s2S\U
XsðÞ2D
fg

n
\
s2U
XsðÞ2D
fg
0:
For any open set UTthe following relationships are true:
sup
t2U
XtðÞ6¼ sup
t2S\U
XtðÞ

0;
inf
t2UXtðÞ6¼ inf
t2S\UXtðÞ

0;
where P(
0
) = 0. If a random process Xis continuous with probability one then it is
separable (Buldygin and Kozachenko, 2000).
We have proved that the process Zis almost surely continuous on [0, 1]. Hence the
process t¼Zte
ZZtis almost surely continuous on [0, 1] too. Therefore it is a separable
process. So we can apply Lemma 4.4 to D
t
.
Let us now reformulate Lemma 4.4 above for our case.
LEMMA 4.6 Let ,,
0
and be as in Theorem 4.3, and let r and be as in Lemma 4.4.
Then for all 2Rand p 2(0, 1) we have
Psup
t20;1½
t
jj
>
!
2exp  þ’0
1p

r10


p1pðÞ
Zp
0
uðÞdu
0
@1
A
2
:ð4:12Þ
Proof: From Proposition 4.2 it follows that for the error process Dwe may take
0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
appr þcut
p:ð4:12Þ
and
hðÞ¼h¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
appr
þcut

qh:ð4:12Þ
In the inequality (4.11) we put ¼min 0;
2

. Since e
0
we have
’0
1p

1pðÞþ’
1p

p’0
1p

392 KOZACHENKO, SOTTINEN AND VASYLYK

So, it follows from the Chebyshev inequality and from (4.11) that for any > 0 we have
Psup
t20;1½
t
jj
>
!
exp  þ’0
1p

2I2
r;
where we have used the denotation
Ir¼r10pðÞþ 
1pðÞpZp2
0
uðÞdu
0
B
@1
C
A:
Since the function t7!re
’tðÞ

is an Orlicz N-function then tðÞ¼re
’tðÞ
ðÞ
tincreases
in tQ0 (cf. Krasnoselskii and Rutitskii, 1958). Therefore, ’
1xðÞðÞ¼
re
x
ðÞ
’1xðÞincreases
in xQ0. Consequently, is a decreasing function. Thus,
pðÞ 1
p1pðÞ
Zp
p2
uðÞdu
and
0pðÞ 0
p1pðÞ
Zp
p2
uðÞdu:
Since 0
1 we have
0pðÞþ 
p1pðÞ
Zp2
0
uðÞdu0


p1pðÞ
Zp
0
uðÞdu:
The claim follows now from Lemma 4.4. Í
Theorem 4.3 follows now by using the YoungYFenchel transformation and then
choosing suitable ,pand rin the inequality (4.12).
Proof of Theorem 4.3: By Proposition 2.4 we know that xy =’(x)+’*(y) when
x=l
j1
(y), where l
j1
is the generalised inverse function of the density lof ’.Since
 ’0
1p

¼0
1p1pðÞ
0’0
1p

we have the equality
 ’0
1p

¼’*1pðÞ
0

SIMULATION OF WEAKLY SELF-SIMILAR STATIONARY INCREMENT SUB’ðÞ-PROCESSES 393

when
0
1p¼l11pðÞ
0

:
So, we choose the following :
¼1p
0
l11pðÞ
0

:
Setting this in the inequality (4.12) we obtain
Psup
t20;1½
t
jj
>
!
2exp ’*1pðÞ
0

r10


p1pðÞ
Zp
0
uðÞdu
0
@1
A
2
¼2exp ’*1pðÞ
0

r10

1pðÞ
0
l11pðÞ
0

1
p1pðÞ
Zp
0
uðÞdu
0
@1
A
2
¼2exp ’*1p
ðÞ
0

r11
p
l11pðÞ
0

Zp
0
uðÞdu
0
@1
A
2
Let us now consider the integral term above. In our case we have
Zp
0
uðÞdu¼Zp
0
rN1uðÞðÞðÞ
’1lnN 1uðÞðÞðÞ
du
R
p
0
r1
21uðÞþ1

’1ln 1
21uðÞþ1

du
¼Zp
0
r1
2

u

1
þ1

’1ln 1
2

u

1
þ1

du:
Now, if the denominator satisfies
’1ln 1
2

u

1
þ1

1
394 KOZACHENKO, SOTTINEN AND VASYLYK

as uep, that is
p
2exp ’1ðÞ
fg
1ðÞ
;ð4:13Þ
then we have
Zp
0
uðÞduZp
0
r1
2

u

1
þ1

du:ð4:13Þ
Let us choose r(x)=x
b
j1, where 0 <b<. Then, by using the estimate above and
the fact that (x+1)
b
jx
b
e1, we obtain
Zp
0
uðÞduZp
0
1
2

u

1


b
du¼b


2b
pðÞ1b

1b

:
Thus, we have obtained the estimate
Psup
t20;1½
t
jj
>
!
2exp ’*1pðÞ
0
no
r1 1
pl11pðÞ
0

b


2b
pðÞ
1b

1b


2
¼2exp ’*1pðÞ
0
no
r1b

pðÞ
b

2b1b

ðÞ
l11pðÞ
0


2
:
For pwe choose
p¼0
ð4:14Þ
(recall that 
0
<) and we obtain the inequality
Psup
t20;1½
t
jj
>
!
2exp ’*
01
no
r1b

0


b

2b1b


l1
01

!
2
¼2exp ’*
01
no
r11
2b1b



0

b

l1
01

!
2
¼2exp ’*
01
no
1
2b1b



0

b

l1
01

þ1
!
2
b
ð4:15Þ
The claim follows from the inequalities (4.13)Y(4.15) and Lemma 4.6. Í
Let us now assume that the constants c
n
and d
n
and the zeros x
n
and y
n
are actually
correctly calculated.
SIMULATION OF WEAKLY SELF-SIMILAR STATIONARY INCREMENT SUB’ðÞ-PROCESSES 395

COROLLARY 4.7 Suppose that there is no approximation error, i.e., 
n
c
=
n
d
=
n
x
=
n
y
=0.
Then the conditions (4.8)Y(4.10) of Theorem 4.3 are satisfied if
Nmax A0


1=H
þ1;A0exp ’1ðÞ
fg
1ðÞ



1=H
þ1;2A0
A

1

()
ð4:16Þ
and
2exp ’*NH
A01

A
ðÞ
b
Nþ1ðÞ
2Hb=
2b1b


A
2b

0NHðÞb

l1Nþ1ðÞ
H
A01
!
þ1
0
@1
A
2
b
;ð4:17Þ
where
A0¼a’ffiffiffiffiffiffiffi
5c
2H
rand A¼21a’ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c
H
r:ð4:17Þ
Proof: Note that now appr ¼appr
¼0.
We shall use the asymptotics xnynnand c2
nd2
nc
n2Hþ1in the expressions for

cut
and cut
.
For 
cut
we get the upper bound
cut ¼a2
’X
1
n¼Nþ1
c2
nþ4d2
n

a2
’X
1
n¼Nþ1
5c
n2Hþ1
5ca2
’X
1
n¼NZ
nþ1
n
dx
x2Hþ1
¼5ca2
’
2HN2H:
For cut
we obtain
cut
¼222a2
’X
1
n¼Nþ1
c2
nx2
nþd2
ny2
n

222a2
’X
1
n¼Nþ1
cnðÞ
2
n2Hþ1þcnðÞ
2
n2Hþ1
!
222a2
’2c2X
1
n¼NZ
nþ1
n
dx
x2HðÞþ1
¼222a2
’c2
HðÞN2HðÞ
:
396 KOZACHENKO, SOTTINEN AND VASYLYK

In the same way we get the lower bounds
cut 5ca2
’
2HNþ1ðÞ
2H;
cut
222a2
’c2
HðÞNþ1ðÞ
2HðÞ
:
Therefore, we have the following asymptotic bounds for 
0
and of Theorem 4.3:
A0
Nþ1ðÞ
H0A0
NH;
A
Nþ1ðÞ
HA
NH:
If
N2A0
A

1

then in Theorem 4.3 we have =
0
. Now we see that the condition (4.8) is satisfied if
NA0


1=H
þ1:
Similarly, (4.9) is satisfied if
NA0exp ’1ðÞ
fg
1ðÞ



1=H
þ1:
Finally, we see that the condition (4.10) is satisfied if (4.17) holds. Í
Theorem 4.3 and Corollary 4.7 are still rather general and not readily useful in
practice. Indeed, there are still the parameters and bone has to optimise. If we choose
a specific form for the function ’we are able to give an applicable version of Corollary
4.7. The next corollary deals with the sub-Gaussian case, i.e., ’(x)=x
2
/2.
COROLLARY 4.8 If the process Z is sub-Gaussian then the conditions (4.16) and (4.17)of
Corollary 4.7 are satisfied if
Nmax a’
ffiffiffiffiffiffiffi
5c
2H
r
!
1=H
þ1;224
H51
H

8
<
:9
=
;ð4:18Þ
SIMULATION OF WEAKLY SELF-SIMILAR STATIONARY INCREMENT SUB’ðÞ-PROCESSES 397

and
2exp 1
2
NH
a’ffiffiffiffiffi
5c
2H
q1
0
B
@1
C
A
2
8
>
<
>
:9
>
=
>
;N14 ; ð4:19Þ
where
¼2222
H458
HH
c

6
H
a’

12
H
:ð4:19Þ
Proof: In the sub-Gaussian case we have ’xðÞ¼x2
2. So,
’*xðÞ¼x2
2and lxðÞ¼’0xðÞ¼x¼l1xðÞ:
Thus, the conditions (4.16) and (4.17) take the form
Nmax A0


1=H
þ1;2A0
A

1

() ð4:20Þ
and
2exp 1
2
NH
A01

2
()
A
ðÞ
b
Nþ1ðÞ
2Hb=
2b1b


A
2b

0NHðÞb

Nþ1ðÞ
H
A01
!
þ1
0
@1
A
2
b
:ð4:21Þ
Let’s take ¼H
2and b¼H
4.
In this case A0¼a’ffiffiffiffiffi
5c
2H
q;A¼AH
2¼a’H
221H
2ffiffiffiffi
2c
H
qand from the inequality (4.20)
we get
Nmax a’
ffiffiffiffiffiffiffi
5c
2H
r
!
1=H
þ1;224
H51
H

8
<
:9
=
;:
Since Nis large we have in (4.21)
A
ðÞ
b
Nþ1ðÞ
2Hb=
2b1b


A
2b

0NHðÞb

Nþ1ðÞ
H
A01
!
þ1
0
@1
A
2
b
N14:
The claim follows. Í
398 KOZACHENKO, SOTTINEN AND VASYLYK

REMARK 4.9 In Corollary 4.8 the condition (4.18) for Nis in closed form. Condition
(4.19) is still implicit, but it may be solved easily using numerical methods. Corollary 4.8
is readily applicable for the fractional Brownian motion. Indeed, in this case a’¼1.
EXAMPLE 4.10 Let
’xðÞ¼
xp
p;x
jj
>1;p>2;
x2
p;x
jj1:
8
>
>
>
<
>
>
>
:
In this case we have:
’*xðÞ¼x2
2;lxðÞ¼’0xðÞ¼x;l1xðÞ¼x
for x2[0, 1] and
’*xðÞ¼xq
q;1
pþ1
q¼1

;lxðÞ¼’0xðÞ¼xp1;l1xðÞ¼x1
p1
for x>1.
Then for 0 
011 the condition (4.10) of Theorem 4.3 takes the form
2exp 1
2

01

2
()
1
2b1b



0

b

01

þ1
!
2
b

and for 
01>1 we have
2exp 1
q

01

q

1
2b1b



0

b

01

1
p1
þ1
!
2
b
:
Acknowledgments
The authors wish to thank Kacha Dzhaparidze, Harry van Zanten, and Esko Valkeila for
fruitful discussions. Sottinen was financed by the European Commission research
training network DYNSTOCH. The authors were partially supported by the European
Union project Tempus Tacis NP 22012-2001 and the NATO Grant PST.CLG.980408.
We also thank the referee for useful comments and for pointing out a serious mistake
in the original manuscript.
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