An approximation to the subfractional Brownian sheet using martingale differences

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R E S E A R C H

Open Access

An approximation to the subfractional

Brownian sheet using martingale differences

Jinhong Zhang, Guangjun Shen

*

_{and Mengyu Li}

*_{Correspondence: [email protected]}

Department of Mathematics, Anhui Normal University, Wuhu, 241000, China

Abstract

In this paper, we obtain an approximation in law of the subfractional Brownian sheet using martingale diﬀerences.

MSC: 60G15; 60G18; 60F05

Keywords: subfractional Brownian sheet; martingale diﬀerences; weak convergence

1 Introduction

As an extension of Brownian motion, Bojdeckiet al.[] introduced and studied the sub-fractional Brownian motion, a class of self-similar Gaussian processes preserving many properties of the fractional Brownian motion (self-similarity, long-range dependence, Hölder paths). However, in comparison with fractional Brownian motion, the subfrac-tional Brownian motion has non-stationary increments. The subfracsubfrac-tional Brownian mo-tion arises from occupamo-tion time ﬂuctuamo-tions of branching particle systems with a Poisson initial condition, and it also appeared independently in a diﬀerent context in Dzhaparidze and Van Zanten [].

Recall that the subfractional Brownian motionSH _{with index}_H_∈_{(, ) is a mean zero}

Gaussian processSH₌_{_SH

t ,t≥}withSH =  and the covariance

RH(t,s)≡E

SH_t SH_s=sH+tH– 

(s+t)H+|t–s|H (.)

for alls,t≥. ForH= /,SHcoincides with the standard Brownian motionW.SHis nei-ther a semimartingale nor a Markov process unlessH= /, so many of the powerful tech-niques from stochastic analysis are not available when dealing withSH. By Dzhaparidze and Van Zanten [], Tudor [], the subfractional Brownian motion has the following in-tegral representation with respect to the standard Brownian motionW:

SH_t =

t



KH(t,s)dWs, t≥, (.)

where

KH(t,s) =

cH√π

H–₍_H_– )

s–H

t

s

u–sH–_du_

(,t)(s), (.)

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with

c_H=( + H)sinπH π , H>

 .

Convergence to subfractional Brownian motion has been studied since the works of Del-gado and Jolis []. Recently, many authors have got some new results on approximations of subfractional Brownian motion. For example, Bardina and Bascompte [] constructed two independent Gaussian processes by using a unique Poisson process. As an application of this result they obtained families of processes that converge in law toward subfractional Brownian motion. Garzónet al.[] proved a strong uniform approximation with a rate of convergence for subfractional Brownian motion by means of transport processes. Harnett and Nualart [] proved a weak convergence of the Stratonovich integral with respect to a class of Gaussian processes which includes subfractional Brownian motion withH=_. Shen and Yan [] obtained an approximation theorem for subfractional Brownian mo-tion using martingale diﬀerences (Nieminen [], Chenet al.[], Wanget al.[] obtained an approximation theorem for fractional Brownian motion, Rosenblatt process, and frac-tional Brownian sheet, respectively, using martingale diﬀerences). Dai [] showed a result of approximation in law to subfractional Brownian motion in the Skorohord topology. The construction of these approximations is based on a sequence of I.I.D. random variables.

There are two possible multidimensional parameter extensions of the subfractional Brownian motion. The first one is Lévy’s subfractional Brownian random field, and the second one is the anisotropic subfractional Brownian random field. In this paper, we will consider the second one in the two-parameter case and call it a subfractional Brownian sheet. This is a centered Gaussian process onR

Sα,β=Sα,β(t,s), (t,s)∈R₊ such that

ESα,β(t,s)Sα,β(u,v)= tH+uH– 

(t+u)H+|t–u|H

× sH+vH– 

(s+v)H+|s–v|H,

whereα,β∈(, ). Forα=β=_,Sα,β _{coincides with the standard Brownian sheet.}

Re-call that a subfractional Brownian sheet with parametersα>_,β>_ admits an integral representation of the form, for (t,s)∈[,T]×[,S],

Sα,β₍_t_,_s_{) =}

t



s



Kα(t,x)Kβ(s,x)B(dx,dx),

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The rest of this paper is organized as follows. Section  contains some preliminaries on the stochastic process in the plane. In Section , we prove the main weak convergence Theorem . using the criterion given by Bickel and Wichura [] to check the tightness of the approximated processes and the convergence of the ﬁnite-dimensional distributions.

2 Preliminaries

In this section, we will use the deﬁnitions and notations introduced in the basic work of Cairoli and Walsh [] on stochastic calculus in the plane. Let (,F,P) be a complete prob-ability space and let{Ft,s; (t,s)∈[,T]×[,S]}be a family of sub-σ-ﬁelds ofFsuch that:

(i) Ft,s⊆Ft,s for anyt≤t,s≤s;

(ii) F,contains all null sets ofF;

(iii) for eachz∈[,T]×[,S],Fz=

z<zFz, wherez= (t,s) <z= (t,s)denotes the

partial order on[,T]×[,S], meaning thatt<tands<s.

Given a real valued processX, deﬁned on [,T]×[,S] with (t,s) < (t,s), we denote by

t,sX(t,s) the increment ofXover the rectangle ((t,s), (t,s)], that is, t,sX

t,s=Xt,s–Xt,s–Xt,s+X(t,s).

Deﬁnition . An integrable processX={X(t,s), (t,s)∈[,T]×[,S]}is called a strong martingale if:

• Xis adapted;

• Xvanishes on the axes;

• E[ t,sX(t,s)|Gt,s] = for any(t,s) < (t,s).

Denote G_i(_,n_j) :=F_i(_,n_n) ∨F_n(n_,_j), where F_i(_,n_n) denotes the σ-ﬁelds generated by ξ_i(_,n_n) and

F(n)

n,j denotes theσ-ﬁelds generated byξ

(n)

n,j fori,j= , , . . . ,nandn≥. Let{ξ(n)}n≥:=

{ξ_i(_,n_j),G_i(_,n_j)}n≥,i,j= , , . . . ,n, be a sequence such that

Eξ_i(_+,n)_j_+|G_i(_,n_j)= 

for alln≥, andξ_k(n_,_l)∈F_i(_,n_n)∧F_n(n_,_j)fori>korj>l. Then we call it a martingale diﬀerences sequence. Obviously, for a martingale diﬀerence sequenceξn,

Xn(i,j) = i

k=

j

l=

ξ_k(n_,_l)

is a strong martingale.

Let be the group of all mappingsλ: [,T]×[,S]→[,T]×[,S] of the formλ(s,t) = (λ(t),λ(s)), where eachλiis continuous and strictly increasing. Denote byD=D([,T]×

[,S]) the Skorohod space of functions on [,T]×[,S] that are continuous from above with limits from below and equippedDwith the metric

d(x,y) :=infminx–yλ,λ:λ∈ ,

wherex–yλ:=sup{|x(t,s) –y(λ(t,s))|: (t,s)∈[,T]×[,S]}and

λ=supλ(t,s) – (t,s): (t,s)∈[,T]×[,S].

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Forn≥, (t,s)∈[,T]×[,S]. Deﬁne now

Bn(t,s) =

[nt]

i= [ns]

j=

ξ_i(_,n_j),

and

Sn(t,s) :=

t



s



K_α(n)(t,x)Kβ(n)(s,x)Bn(dx,dx)

=n

[nt]

i= [ns]

j=

ξ_i(_,n_j)

i n

i–

n

j n

j–

n

Kα

[nt]

n ,x

·Kβ

[ns]

n ,x

dxdx, (.)

where the sequenceK_H(n)(t,v) is an approximation ofKH(t,v) and

K_H(n)(t,x) =n x

x–_n

KH

[nt]

n ,y

dy, n= , , . . .

and [x] denotes the greatest integer not exceedingx.

It is well known that if the martingale diﬀerences sequenceξ(n)_{satisﬁes the following}

condition:

[nt]

i= [ns]

j=

ξ_i(_,n_j)→t·s

in the sense ofL, thenBn(t,s) converges weakly to the Brownian sheetB(t,s) inDasn

tends to inﬁnity (see, for instance, Morkvenas []).

3 Main results

In this section, we will extend the result of Morkvenas [] to the subfractional Brownian sheet withα,β>_ in the following theorem.

Theorem . Letα>_,β>_,and{ξ_i(_,n_j),i,j= , , . . . ,n}be a square integrable martingale diﬀerences sequence such that for all≤i,j≤n

lim

n→∞nξ

(n)

i,j =  a.s. (.)

and

max

≤i,j≤n

ξ_i(_,n_j)≤C

n a.s. (.)

for some C> .Then{Sn}deﬁned in(.)converges weakly to the subfractional Brownian sheet Sα,β_{in the Skorohod space}_D_{as n tends to inﬁnity}_.

In order to prove Theorem ., we have to check the tightness of processSnand the

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Lemma . Let Sn(t,s)be the family of processes deﬁned by(.).Then for any(t,s) < (t,s), there exists a constant C such that

sup

n

E t,sSn

t,s≤Ct–tαs–sβ.

Proof Notice that

t,sSn

t,s=

t



s



Kα(n)

t,x

–Kα(n)(t,x)

×K_β(n)s,x

–K_β(n)(s,x)

Bn(dx,dx)

=

[nt]

i= [ns]

j= n i n i– n j n j– n Kα

[nt]

n ,x

–Kα

[nt]

n ,x

×

Kβ

[ns]

n ,x

–Kβ

[ns]

n ,x

ξ_i(_,n_j)dxdx.

By (.) and Eξ_i(_,n_j)ξ_k(_,n_l)= , for some i=k or j=l [in fact, we can supposei>k, then

Eξ_i(_,n_j)ξ_k(n_,_l)=E(ξ_k(n_,_l)E(ξ_i(_,n_j)|G_i(_–,n)_j_–)) = , sinceξ_k(n_,_l)∈F_i(_,n_n)∧F_n(n_,_j)]. We have

E t,sSn

t,s =E _[_nt_]

i= [ns]

[nt]

n ,x

–Kα

[nt]

n ,x

×

Kβ

[ns]

n ,x

–Kβ

[ns]

n ,x

ξ_i(_,n_j)dxdx



≤C

[nt]

i= n i n i– n Kα

[nt]

n ,x

–Kα

[nt]

n ,x

dx



×

[ns]

j= n j n j– n Kβ

[ns]

n ,x

–Kβ

[ns]

n ,x dx  ≤C t  Kα

[nt]

n ,x

–Kα

[nt]

n ,x

 dx × s  Kβ

[ns]

n ,x

–Kβ

[ns]

n ,x



dx

=CESH[nt]

n

–SH[nt]

n



ESH[ns]

n

–SH[ns]

n



≤C[nt

_{] – [}_nt_]

n

α[ns] – [_n ns]β.

For any  <t<t, and _<α< , we see that ifnt–nt≥, then|[nt]–[_nnt]|α_{≤ |}_(_t_–_t₎_|α_.

Conversely, ifnt–nt< , then eithertortbelong to a same subinterval [m_n,m_n+) for some integerm, which implies|[nt]–[_nnt]|α_{= . So we get}

[nt] – [_n nt]α≤t–tα.

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Lemma . Let Sn(t,s)be the family of processes deﬁned by(.).Then for any(t,s) < (t,s), there exists a constant C such that

sup

n

E t,sSn

t,s≤Ct–tαs–sβ.

Proof We have

E t,sSn

t,s=E _[_nt_]

i= [ns]

[nt]

n ,x

–Kα

[nt]

n ,x

×

Kβ

[ns]

n ,x

–Kβ

[ns]

n ,x

ξ_i(_,n_j)dxdx



≤C

[nt]

i= [ns]

j= [nt]

k= [ns]

l= n i n i– n Kα

[nt]

n ,x

–Kα

[nt]

n ,x dx  × j n j– n Kβ

[ns]

n ,x

–Kβ

[ns]

n ,x dx  × k n k– n Kα

[nt]

n ,x

–Kα

[nt]

n ,x dx  × l n l– n Kβ

[ns]

n ,x

–Kβ

[ns]

n ,x

dx



=C _[_nt_]

i= n i n i– n Kα

[nt]

n ,x

–Kα

[nt]

n ,x

dx



×

_[_ns_]

i= n j n j– n Kβ

[ns]

n ,x

–Kβ

[ns]

n ,x

dx



≤C _[_nt_]

i= i n i– n Kα

[nt]

n ,x

–Kα

[nt]

n ,x



dx



×

_[_ns_]

j= j n j– n Kβ

[ns]

n ,x

–Kβ

[ns]

n ,x



dx



≤C[nt

_{] – [}_nt_]

n

α[ns] – [_n ns]β

≤Ct–tαs–sβ.

Lemma . Let _ <α,β< , (tk,sk), (tl,sl)∈[,T]×[,S],and{ξi(,nj),i,j= , , . . . ,n}be a martingale diﬀerences sequence satisfying(.)and(.).Then

n

[nT]

i= [nS]

j= i n i– n j n j– n Kα

[ntk] n ,x

Kβ

[nsk] n ,x

dxdx

× i n i– n j n j– n Kα

[ntl] n ,x

Kβ

[nsl] n ,x

dxdx

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converges to

T



Kα(tk,x)Kα(tl,x)dx

S



Kβ(sk,x)Kβ(tl,x)dx

as n tends to inﬁnity.

Proof The proof follows from a similar discussion of Lemma . in Shen and Yan [].

Proof of Theorem. Note that the processesSnare null on the axes, and that the tightness

of the laws of the familySnfollows from Lemma .. Then, in order to prove Theorem .,

by the criterion given by Bickel and Wichura [] it suﬃces to show: the family of process

Sn(t,s) converges, asntends to inﬁnity, to the subfractional Brownian sheet in the sense

of ﬁnite-dimensional distribution.

Letα,α, . . . ,αd∈Rand (t,s), . . . , (td,sd)∈[,T]×[,S]. We want to show thatXn,

Xn:= d

m=

αmSn(tm,sm),

converges in distribution, asntends to inﬁnity, to a normal random variable with zero mean and variance

E

_d

m=

αmSα,β(tm,sm)



.

Indeed, the zero mean is trivial. Let us writeXnas

Xn=

[nT]

i= [nS]

j=

nξ_i(_,n_j)

d

m=

αm

i n

i–

n

Kα

[ntm] n ,x

dx

j n

j–

n

Kβ

[nsm] n ,x

dx

:=

[nT]

i= [nS]

j=

X_i(_,n_j).

Then

[nT]

i= [nS]

j=

X_i(_,n_j)=

d

m,h=

αmαhn

[nT]

i= [nS]

j=

i n

i–

n

j n

j–

n

Kα

[ntm] n ,x

Kβ

[nsm] n ,x

dxdx

×

i n

i–

n

j n

j–

n

Kα

[nth] n ,x

Kβ

[nsh] n ,x

dxdx

ξ_i(_,n_j).

By Lemma ., the above equation converges to

d

m,h=

αmαh

T



Kα(tm,x)Kα(th,x)dx

S



Kβ(sm,x)Kβ(sh,x)dx

=E

_d

m=

αmSα,β(tm,sm)



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sinceKα(t,s) =  fors≥t. Therefore, in order to end the proof we just need to prove the

following Lindeberg condition holds: for anyε> ,

[nT]

i= [nS]

j=

EX_i(_,n_j)I

(|X_i(_,n_j)|>ε)|G

n i–,j–

P

→.

Consider the set

X(_i_,n_j)>ε=X_i(_,n_j)>ε.

We get an upper bound toX(_i_,n_j)by noticing thatKH(t,u) is increasing with respect totand

decreasing with respect tou,

X_i(_,n_j)=

nξ_i(_,n_j)

d

m=

αm

i n

i–

n

j n

j–

n

Kα

[ntm] n ,x

Kβ

[nsm] n ,x

dxdx



≤nξ_i(_,n_j)

_d

m=

αm

 i

n

i–

n

Kα(T,x)dx



·

j n

j–

n

Kβ(S,x)dx



≤nξ_i(_,n_j)A i

n

i–

n

Kα(T,x)dx·

j n

j–

n

Kβ(S,x)dx

≤nξ_i(_,n_j)A:=nξ_i(_,n_j)Aδn,

whereA:= (d_m_=αm), and

δn:=



n



Kα(T,x)dx·



n



Kβ(S,x)dx.

Thus, we get

_X(n)

i,j>ε

⊂nξ_i(_,_jn)Aδn>ε. (.)

Using the inclusion (.) and the Cauchy-Schwartz inequality we get, a.s.,

EX_i(_,n_j)I

(|X(_i_,n_j)|>ε)|G

n i–,j–

≤Enξ_i(_,_jn)AδnI

(n₍_ξ(n)

i,j )Aδn>ε)|G n i–,j–

≤CAδnEI₍_n₍_ξ(n)

i,j )Aδn>ε)|G n i–,j–

.

Hence, for some constantK> 

[nT]

i= [nS]

j=

EX_i(_,n_j)I₍_|_X(n)

i,j|>ε)|G n i–,j–

≤

[nT]

i= [nS]

j=

CAδnEI₍_n₍_ξ(n)

i,j )Aδn>ε)|G n i–,j–

a.s.

≤CAδn

[nT]

i= [nS]

j=

E[I(K_A_δn_>_ε₎]→, n→ ∞,

becauseδn_→_{ implies}_I

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The main idea of this paper was proposed by the corresponding author GJS. JHZ, GJS, and MYL prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the ﬁnal manuscript.

Acknowledgements

The authors would like to thank the anonymous highly diligent referees for their careful reading of the manuscript and helpful comments. This work was supported by National Natural Science Foundation of China (11271020).

Received: 8 October 2014 Accepted: 7 March 2015 References

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