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Simulation of Weakly Self-Similar Stationary

Increment Sub’’’’’’ððððÞÞÞÞÞð -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’ðÞ-processes that are weakly selfsimilar with stationary increments

in the sense that they have the covariance function

Rt;sðÞ¼

1

2t2Hþs2Hts

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=2ðÞ-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

2t2Hþs2Hts

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

t0¼

daHZat

t0

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

1cos 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’’’’’’’ððððÞÞÞÞ

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;x0;

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*xðÞ¼xy0uy

0

ðÞif y0¼l1xðÞ:

Here l

j1

is the generalised inverse function of l, i.e.,

l1xðÞ:¼sup v0:lvðÞx

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

jj1;

(p>1;

’xðÞ¼

e

2

2

x2if x

jj2

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 ’aðÞ

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 ’aðÞ

fg

;2R

fg

:

Moreover, for any 2Rwe have

Eexp

fg

exp ’

’ðÞ

;E2

1

22CðÞ

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., ªªeca.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.,

JxðÞ¼X

1

n¼0

1ðÞ

nx=2ðÞ

þ2n

nþ1ðÞþnþ1ðÞ

:

Here x>0,mj1, j2, . . . and Gdenotes the Euler Gamma function

zðÞ¼Z1

0

tz1etdt:

It is well-known that for >j1 the Bessel function Jhas 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

dn1cos 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

cn¼Hﬃﬃﬃﬃﬃ

2c

p

xHþ1

nJ1Hxn

ðÞ

;n¼1;2;:::; ð3:1Þ

dn¼Hﬃﬃﬃﬃﬃ

2c

p

yHþ1

nJHyn

ðÞ

;n¼1;2;:::; ð3:2Þ

c¼2Hþ1ðÞsin 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 xntðÞnþX

1

n¼1

dn1cos yntðÞðÞn;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 þn

ðÞðÞ

1þ<1

for some >0then the series

X

1

n¼1

ancos ntðÞnand X

1

n¼1

ansin ntðÞn

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

õnas nYV. Also from (Watson, 1944),

p. 200, we have the following asymptotic relation for the Bessel function Jfor >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

1Hxn

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 yntðÞnand almost sure convergence of the series

P1

n¼1dnnimply that P1

n¼1dn1cos yntðÞðÞnalso 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

EZtZs

jj

n¼EZts

jj

n¼ts

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

Zte

ZZt

>

!

:

A natural model for Z, deﬁned by the expansion (3.4), would be the truncated series

X

N

n¼1

cnsin xntðÞnþdn1cos yntðÞðÞn

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

ccncn

jj

c

n;

je

ddndnjd

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

xxnxn

jj

x

n;

e

yynyn

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

xxntðÞnþe

ddn1cos e

yyntðÞðÞn

:ð4:1Þ

The error in the simulation (model) is

t:¼Zte

Zt

Zt

¼X

N

n¼1

cnsin xntðÞ

e

ccnsin e

xn

xntðÞðÞnþdn1cos yntðÞðÞ

e

ddn1cos e

yyntðÞðÞ

n

no

þX

1

n¼Nþ1

cnsin xntðÞnþdn1cos yntðÞðÞn

fg

¼:

appr

tþcut

t:

In order to bound the error Din C([0, 1]) we need estimates for ’t

ðÞand

’ts

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

cnx

nþc

n

2þdny

nþ2d

n

2

no

:

Let 2(0, H)and denote

cut

:¼222a2

’X

1

n¼Nþ1

c2

nx2

nþd2

ny2

n

;

appr

:¼232a2

’X

N

n¼1

x2

nc

n

2þy2

nd

n

2þ232e

cn

cn

ðÞ

2x

n

2xnþ~

xxn

ðÞ

2

22þ1

(

þe

ddn

2y

n

2ynþ~

yyn

2

22þ1!):

386 KOZACHENKO, SOTTINEN AND VASYLYK

Then we have for all s, t 2[0, 1]

2

’t

ðÞappr þcut ;ð4:2Þ

2

’ts

ðÞappr

þcut

ts

jj

2:ð4:3Þ

Proof: By Proposition 2.12 we know that

2

’t

ðÞ2

’appr

t

ðÞþ2

’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

22

’n

ðÞ

þX

N

n¼1

dn1cos yntðÞðÞ

e

ddn1cos e

yyntðÞðÞ

22

’n

ðÞ

¼a2

’X

N

n¼1

cnsin xntðÞsin e

xxntðÞðÞþcne

ccn

ðÞsin e

xxntðÞðÞ

2

(

þX

N

n¼1

dncos e

yyntðÞcos yntðÞðÞþdne

ddn

1cos e

yyntðÞðÞ

2

a2

’X

N

n¼1

cnxne

xxn

jj

tþcne

ccn

ðÞsin e

xxntðÞ

jj

ðÞ

2

(

þX

N

n¼1

dne

yynyn

jjtþdne

ddn

1cos e

yyntðÞðÞ

2

a2

’X

N

n¼1

cnx

nþc

n

2þdny

nþ2d

n

2

no

:

¼appr:

Similarly we obtain

2

’cut

t

a2

’X

1

n¼Nþ1

c2

nþ4d2

n

¼cut: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

’ts

ðÞ. For the Bcut-off^part

we have

2

’cut

tcut

s

¼2

’X

1

n¼Nþ1

cnsin xntðÞsin xnsðÞðÞnþX

1

n¼Nþ1

dncos ynsðÞcos yntðÞðÞn

!

221ðÞ

a2

’X

1

n¼Nþ1

c2

nxnts

jj

ðÞ

2þd2

nynts

jj

ðÞ

2

¼221ðÞ

a2

’X

1

n¼Nþ1

c2

nx2

nþd2

ny2

n

ts

jj

2

¼cut

ts

jj

2:ð4:5Þ

Due to the asymptotics (3.5) and x

n

õy

n

õnthe series in (4.5) converge if <H.

For the Bapproximating part^we have

2

’appr

tappr

s

a2

’X

N

n¼1

cnsin xntðÞsin xnsðÞðÞ

e

ccnsin e

xxntðÞsin e

xxnsðÞðÞðÞ

2

(

þX

N

n¼1

dncos ynsðÞcos yntðÞðÞ

e

ddncos e

yynsðÞcos e

yyntðÞðÞ

2):ð4:6Þ

By using the relations

ab cdjjeajjbdjjþacjjdjj;

aþbðÞ

2e2a2þ2b2;

sin xsin y¼2cos xþy

2sin xy

2

cos xcos y¼2sin xþy

2sin yx

2

we obtain for the summand in (4.6) that

cnsin xntðÞsin xnsðÞðÞ

e

ccnsin e

xxntðÞsin e

xxnsðÞðÞ

jj

sin xntðÞsin xnsðÞ

jj

cne

ccn

jj

þsin xntðÞsin xnsðÞsin e

xxntðÞþsin e

xxnsðÞ

jj

e

ccn

jj

;

sin xntðÞsin xnsðÞ

jj

¼2sin xntsðÞ

2

cos xntþsðÞ

2

2xntsðÞ

2

388 KOZACHENKO, SOTTINEN AND VASYLYK

and

sin xntðÞsin xnsðÞsin e

xxntðÞþsin e

xxnsðÞ

jj

¼sin xntðÞsin e

xxntðÞðÞsin xnsðÞsin e

xxnsðÞðÞ

jj

¼2sin xne

xxn

ðÞt

2

cos xnþe

xxn

ðÞt

2

:

2sin xne

xxn

ðÞs

2

cos xnþe

xxn

ðÞs

2

2 sin xne

xxn

ðÞt

2

cos xnþe

xxn

ðÞt

2

cos ðxne

xxnÞs

2!

þsin xne

xxn

ðÞt

2sin xne

xxn

ðÞs

2

cos xnþe

xxn

ðÞs

2

¼2sin xne

xxn

ðÞt

2

2sin xnþe

xxn

ðÞtþsðÞ

4

sin xnþe

xxn

ðÞstðÞ

4

½

þ2cos xnþe

xxn

ðÞtþsðÞ

4

sin xne

xxn

ðÞðtsÞ

4cos xnþe

xxn

ðÞs

2

4

xne

xxn

ðÞt

2

xnþe

xxn

ðÞðstÞ

4

þ

xne

xxn

ðÞtsðÞ

4

222x

n

xnþe

xxn

ðÞ

2þ1

ts

jj

:

Hence

cnsin xntsin xnse

ccnsin e

xxntsin e

xxns2

2xntsðÞ

2

c

nþ222x

n

xnþ~

xxn

ðÞ

2þ1

ts

jj

e

ccn

jj

2

22

xntsðÞ

2

c

n

2þ22

22x

n

xnþ~

xxn

ðÞ

2þ1

ts

jj

e

ccn

jj

2

¼ts

jj

2232x2

nc

n

2þ254e

ccn

ðÞ

2x

n

2xnþ~

xxn

ðÞ

2þ1

2

ts

jj

2232x2

nc

n

2þ254e

ccn

ðÞ

2x

n

22xnþ~

xxn

ðÞ

2

2þ2

¼tsjj

2232x2

nc

n

2þ264e

ccn

ðÞ

2x

n

2xnþ~xxn

ðÞ

2

22þ1

¼232ts

jj

2x2a

nc

n

2þ232e

ccn

ðÞ

2x

n

2xnþ~xxn

ðÞ

2

22þ1

:

SIMULATION OF WEAKLY SELF-SIMILAR STATIONARY INCREMENT SUB’ðÞ-PROCESSES 389

In the same way

dncos ynsðÞcos yntðÞðÞ

e

dncos e

yynsðÞcos e

yyntðÞðÞ

2

cos ynsðÞcos yntðÞ

jj

dne

dn

þcos ynsðÞcos yntðÞcos e

yynsðÞþcos e

yyntðÞ

jj

e

dn

2

21yn

jj

ts

jj

d

nþ222y

n

ts

jj

ynþ~yn

ðÞ

2þ1

e

ddn

2

22

1yn

jj

ts

jj

d

n

2þ22

22y

n

ts

jj

ynþ~yn

ðÞ

2þ1

e

dn

2

ts

jj

2232y2

nd

n

2þ254e

dn

2y

n

22ynþ~yn

ðÞ

2

22þ2

¼232ts

jj

2y2

nd

n

þ232e

dn

2y

n

2ynþ~yn

ðÞ

2

22þ1

:

Thus, we obtain following estimate

2

’appr

tappr

s

a2

’P

N

n¼1

232

ts

jj

2x2

nc

n

2þ232e

ccn

ðÞ

2x

n

2xnþ~xn

ðÞ

2

22þ1

þP

N

n¼1

232ts

jj

2y2

nd

n

2þ232e

ddn

2y

n

2ynþ~yn

ðÞ

2

22þ1

¼232a2

’ts

jj

2P

N

n¼1

x2

n

c

n

2þy2

nd

n

2

þ232~

ccn

ðÞ

2x

n

2xnþ~xn

ðÞ

2

22þ1

þe

ddn

2y

n

2ynþ~yn

ðÞ

2

22þ1

¼appr

tsjj

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 þcut

p;

¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

appr

þcut

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

<

2exp ’1ðÞ

fg

1ðÞ

;ð4:9Þ

2 exp ’*

01

1

2b1b

0

b

l1

01

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

ts

jj

h

’XtXs

ðÞhðÞ:

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

rN1uðÞðÞðÞ

’1ln N1uðÞðÞðÞ

;

and N(")is the minimum number of closed intervals of the radius "that is needed to

cover the interval [0,1] (note that N "ðÞ1

2"þ1).

Then for all 2Rand p 2(0,1) we have

Eexp sup

t20;1½

Xt

jj

()

2exp ’0

1p

1pðÞþ’

1p

p

r10pðÞþ

1pðÞpZp2

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 STand 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 UTthe 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 t¼Zte

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

1p

r10

p1pðÞ

Zp

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 þcut

p:ð4:12Þ

and

hðÞ¼h¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

appr

þcut

qh:ð4:12Þ

In the inequality (4.11) we put ¼min 0;

2

. Since e

0

we have

’0

1p

1pðÞþ’

1p

p’0

1p

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

1p

2I2

r;

where we have used the denotation

Ir¼r10pðÞþ

1pðÞpZp2

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,

pðÞ 1

p1pðÞ

Zp

p2

uðÞdu

and

0pðÞ 0

p1pðÞ

Zp

p2

uðÞdu:

Since 0

1 we have

0pðÞþ

p1pðÞ

Zp2

0

uðÞdu0

p1pðÞ

Zp

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

1p

¼0

1p1pðÞ

0’0

1p

we have the equality

’0

1p

¼’*1pðÞ

0

SIMULATION OF WEAKLY SELF-SIMILAR STATIONARY INCREMENT SUB’ðÞ-PROCESSES 393

when

0

1p¼l11pðÞ

0

:

So, we choose the following :

¼1p

0

l11pðÞ

0

:

Setting this in the inequality (4.12) we obtain

Psup

t20;1½

t

jj

>

!

2exp ’*1pðÞ

0

r10

p1pðÞ

Zp

0

uðÞdu

0

@1

A

2

¼2exp ’*1pðÞ

0

r10

1pðÞ

0

l11pðÞ

0

1

p1pðÞ

Zp

0

uðÞdu

0

@1

A

2

¼2exp ’*1p

ðÞ

0

r11

p

l11pðÞ

0

Zp

0

uðÞdu

0

@1

A

2

Let us now consider the integral term above. In our case we have

Zp

0

uðÞdu¼Zp

0

rN1uðÞðÞðÞ

’1lnN 1uðÞðÞðÞ

du

R

p

0

r1

21uðÞþ1

’1ln 1

21uðÞþ1

du

¼Zp

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 uep, that is

p

2exp ’1ðÞ

fg

1ðÞ

;ð4:13Þ

then we have

Zp

0

uðÞduZp

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

Zp

0

uðÞduZp

0

1

2

u

1

b

du¼b

2b

pðÞ1b

1b

:

Thus, we have obtained the estimate

Psup

t20;1½

t

jj

>

!

2exp ’*1pðÞ

0

no

r1 1

pl11pðÞ

0

b

2b

pðÞ

1b

1b

2

¼2exp ’*1pðÞ

0

no

r1b

pðÞ

b

2b1b

ðÞ

l11pðÞ

0

2

:

For pwe choose

p¼0

ð4:14Þ

(recall that

0

<) and we obtain the inequality

Psup

t20;1½

t

jj

>

!

2exp ’*

01

no

r1b

0

b

2b1b

l1

01

!

2

¼2exp ’*

01

no

r11

2b1b

0

b

l1

01

!

2

¼2exp ’*

01

no

1

2b1b

0

b

l1

01

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

Nmax A0

1=H

þ1;A0exp ’1ðÞ

fg

1ðÞ

1=H

þ1;2A0

A

1

()

ð4:16Þ

and

2exp ’*NH

A01

A

ðÞ

b

Nþ1ðÞ

2Hb=

2b1b

A

2b

0NHðÞb

l1Nþ1ðÞ

H

A01

!

þ1

0

@1

A

2

b

;ð4:17Þ

where

A0¼a’ﬃﬃﬃﬃﬃﬃﬃ

5c

2H

rand A¼21a’ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

c

H

r:ð4:17Þ

Proof: Note that now appr ¼appr

¼0.

We shall use the asymptotics xnynnand c2

nd2

nc

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

¼222a2

’X

1

n¼Nþ1

c2

nx2

nþd2

ny2

n

222a2

’X

1

n¼Nþ1

cnðÞ

2

n2Hþ1þcnðÞ

2

n2Hþ1

!

222a2

’2c2X

1

n¼NZ

nþ1

n

dx

x2HðÞþ1

¼222a2

’c2

HðÞN2HðÞ

:

396 KOZACHENKO, SOTTINEN AND VASYLYK

In the same way we get the lower bounds

cut 5ca2

’

2HNþ1ðÞ

2H;

cut

222a2

’c2

HðÞNþ1ðÞ

2HðÞ

:

Therefore, we have the following asymptotic bounds for

0

and of Theorem 4.3:

A0

Nþ1ðÞ

H0A0

NH;

A

Nþ1ðÞ

HA

NH:

If

N2A0

A

1

then in Theorem 4.3 we have =

0

. Now we see that the condition (4.8) is satisﬁed if

NA0

1=H

þ1:

Similarly, (4.9) is satisﬁed if

NA0exp ’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

Nmax a’

ﬃﬃﬃﬃﬃﬃﬃ

5c

2H

r

!

1=H

þ1;224

H51

H

8

<

:9

=

;ð4:18Þ

SIMULATION OF WEAKLY SELF-SIMILAR STATIONARY INCREMENT SUB’ðÞ-PROCESSES 397

and

2exp 1

2

NH

a’ﬃﬃﬃﬃﬃ

5c

2H

q1

0

B

@1

C

A

2

8

>

<

>

:9

>

=

>

;N14 ; ð4:19Þ

where

¼2222

H458

HH

c

6

H

a’

12

H

:ð4:19Þ

Proof: In the sub-Gaussian case we have ’xðÞ¼x2

2. So,

’*xðÞ¼x2

2and lxðÞ¼’0xðÞ¼x¼l1xðÞ:

Thus, the conditions (4.16) and (4.17) take the form

Nmax A0

1=H

þ1;2A0

A

1

() ð4:20Þ

and

2exp 1

2

NH

A01

2

()

A

ðÞ

b

Nþ1ðÞ

2Hb=

2b1b

A

2b

0NHðÞb

Nþ1ðÞ

H

A01

!

þ1

0

@1

A

2

b

:ð4:21Þ

Let’s take ¼H

2and b¼H

4.

In this case A0¼a’ﬃﬃﬃﬃﬃ

5c

2H

q;A¼AH

2¼a’H

221H

2ﬃﬃﬃﬃ

2c

H

qand from the inequality (4.20)

we get

Nmax a’

ﬃﬃﬃﬃﬃﬃﬃ

5c

2H

r

!

1=H

þ1;224

H51

H

8

<

:9

=

;:

Since Nis large we have in (4.21)

A

ðÞ

b

Nþ1ðÞ

2Hb=

2b1b

A

2b

0NHðÞb

Nþ1ðÞ

H

A01

!

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

jj1:

8

>

>

>

<

>

>

>

:

In this case we have:

’*xðÞ¼x2

2;lxðÞ¼’0xðÞ¼x;l1xðÞ¼x

for x2[0, 1] and

’*xðÞ¼xq

q;1

pþ1

q¼1

;lxðÞ¼’0xðÞ¼xp1;l1xðÞ¼x1

p1

for x>1.

Then for 0

011 the condition (4.10) of Theorem 4.3 takes the form

2exp 1

2

01

2

()

1

2b1b

0

b

01

þ1

!

2

b

and for

01>1 we have

2exp 1

q

01

q

1

2b1b

0

b

01

1

p1

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.

References

J. Beran, Statistics for Long-Memory Processes, Chapman and Hall: New York, 1994.

V. V. Buldygin and Y. V. Kozachenko, Metric Characterization of Random Variables and Random Processes,

American Mathematical Society, Providence: RI, 2000.

SIMULATION OF WEAKLY SELF-SIMILAR STATIONARY INCREMENT SUB’ðÞ-PROCESSES 399

L. Decreusefond and A. S. U

¨stu¨ nel, BStochastic analysis of the fractional Brownian motion,^Potential analysis

vol. 10 (2) pp. 177Y214, 1999.

P. Doukhan, G. Oppenheim, and M. Taqqu (eds.), Theory and Applications of Long-Range Dependence,

Birkha¨ user Boston, Inc.: Boston, MA, 2003.

K. O. Dzhaparidze and J. H. van Zanten, BA series expansion of fractional Brownian motion,^Probability

Theory and Related Fields vol. 130 pp. 39Y55, 2004.

P. Embrechts and M. Maejima, Selfsimilar Processes, Princeton University Press: Princeton, 2002.

G. A. Hunt, BRandom Fourier transforms,^Transactions of the American Mathematical Society vol. 71

pp. 38Y69, 1951.

A. N. Kolmogorov, BWienersche Spiralen und einige andere interessante Kurven in Hilbertschen Raum,^

Comptes Rendus (Doklady) Acad. Sci. USSR (N.S.) vol. 26 pp. 115Y118, 1940.

Y. V. Kozachenko and O. I. Vasilik, BOn the distribution of suprema of Sub

’

() random processes,^Theory of

Stochastic Proc. vol. 4 (20) pp. 1Y2, 1998, pp. 147Y160.

M. A. Krasnoselskii and Y. B. Rutitskii, Convex Functions in the Orlicz Spaces, Fizmatiz: Moscow, 1958.

B. Mandelbrot and J. Van Ness, BFractional Brownian motions, fractional noises and applications,^SIAM

Review vol. 10 pp. 422Y437, 1968.

G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian random processes, Chapman and Hall: New York,

1994.

G. N. Watson, A Treatise of the Theory of Bessel Functions, Cambridge University Press: Cambridge, England,

1944.

400 KOZACHENKO, SOTTINEN AND VASYLYK