VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA34 10.15559/15-VMSTA34 Research Article Tempered Hermite process SabzikarFarzadsabzikar@iastate.edu Department of Statistics, Iowa State University, Ames, IA 50010, USA 2015 259201524327341 972015 792015 1192015 © 2015 The Author(s). Published by VTeX2015 Open access article under the CC BY license.

A tempered Hermite process modifies the power law kernel in the time domain representation of a Hermite process by multiplying an exponential tempering factor λ>0 0$]]> such that the process is well defined for Hurst parameter H>12 \frac{1}{2}$]]>. A tempered Hermite process is the weak convergence limit of a certain discrete chaos process.

 Discrete chaos limit theorem Wiener–Itô integral Fourier transform 60F17 60G23 60G20 Introduction

The Hermite processes of order k=1,2,… are defined as multiple Wiener–Itô integrals ZHk(t):=∫Rk′∫0t( ∏i=1k(s−yi)+d−1)dsB(dy1)…B(dyk), where d=12−1−Hk∈(12−12k,12) and 12<H<1 (the prime ′ on the integral sign shows that one does not integrate on the diagonals xi=xj , i≠j ). They are self-similar processes with stationary increments (see [8, 26]).

In this paper, we introduce a new class of stochastic processes, which we call tempered Hermite processes. Tempered Hermite processes modify the kernel of ZHk by multiplying an exponential tempering factor λ>0 0$]]> such that they are well defined for Hurst parameter H>12 \frac{1}{2}$]]>. Tempered Hermite processes are not self-similar processes, but they have a scaling property, involving both the time scale and the tempering parameter. The scaling property enable us to show that the tempered Hermite processes are the weak convergence limits of certain discrete chaos processes.

The paper is organized as follows. In Section 2, we define tempered Hermite processes and derive some their basic properties. In Section 3, we present our main result on the weak convergence to tempered Hermite processes.

 Tempered Hermite process

Let B={B(t),t∈R} be a real-valued Brownian motion on the real line, a process with stationary independent increments such that B(t) has a Gaussian distribution with mean zero and variance |t| for all t∈R . Then the Wiener–Itô integrals Ik(f):=∫Rk′f(x1,…,xk)B(dx1)…B(dxk) are defined for all functions f∈L2(Rk) . The prime ′ on the integral sign shows that one does not integrate on the diagonals xi=xj , i≠j . See, for example, [12, Chapter 4].

Let H>12 \frac{1}{2}$]]> and λ>0 0$]]>. The process ZH,λk(t):=∫Rk′∫0t ∏i=1k((s−yi)+d−1e−λ(s−yi)+)dsB(dy1)…B(dyk), where (x)+=xI(x>0) 0)$]]> and d=12−1−Hk∈(12−12k,∞) , is called a tempered Hermite process of order k.

The next lemma shows that ZH,λk(t) , given by (2), is well defined for any t≥0 .

The function ht(y1,…,yk):=∫0t ∏i=1k(s−yi)+d−1e−λ(s−yi)+ds is well defined in L2(Rk) for any H>12 \frac{1}{2}$]]> and λ>0 0$]]>.

To show that ht(y1,…,yk) is in L2(Rk) , we write ∫Rkht(y1,…,yk)2dy1…dyk=∫Rk[∫0t∫0t ∏i=1k(s1−yi)+d−1e−λ(s1−yi)+(s2−yi)+d−1×e−λ(s2−yi)+ds1ds2]dy1…dyk=2∫0tds1∫s1tds2[∫Rk ∏i=1k(s1−yi)+d−1e−λ(s1−yi)+(s2−yi)+d−1×e−λ(s2−yi)+dy1…dyk]=2∫0tds∫0t−sdu[∫R+k ∏i=1kwid−1e−λwi(wi+u)d−1e−λ(wi+u)dw1…dwk](s=s1,u=s2−s1,wi=s1−yi)=2∫0tds∫0t−se−λukdu[∫R+wd−1(w+u)d−1e−2λwdw]k=2∫0tds∫0t−se−λukuk(2d−1)du[∫R+xd−1(x+1)d−1e−2λuxdx]k=2∫0tds∫0t−se−λukuk(2d−1)du[Γ(d)π(12λu)d−12eλuK12−d(λu)]k=2[Γ(d)π(2λ)d−12]k∫0tds∫0t−s[ud−12K12−d(λu)]kdu=2[Γ(d)π2d−12λ2d−1]k∫0tds∫0λ(t−s)[zd−12K12−d(z)]kdz, where we applied the standard integral formula [13, p. 344] ∫0∞xν−1(x+β)ν−1e−μxdx=1π(βμ)ν−12eβμ2Γ(ν)K12−ν(βμ2) for |argβ|<π , Re μ>0 0$]]>, Re ν>0 0$]]>. Here Kν(x) is the modified Bessel function of the second kind (see [1, Chapter 9]). Next, we need to show that ∫0tds∫0λ(t−s)[zd−12K12−d(z)]kdz is finite for d>12−12k \frac{1}{2}-\frac{1}{2k}$]]> (equivalently, for H>12 \frac{1}{2}$]]>). First, suppose 12−12k<d<12 (or 12<H<1 ). Since kν(z)<z−ν2ν−1Γ(ν) for z>0 0$]]> (Theorem 3.1 in [11]), we have ∫0tds∫0λ(t−s)[zd−12K12−d(z)]kdz≤[2−(12+d)Γ(12−d)]k∫0tds∫0λ(t−s)zk(2d−1)dz=[λ2d−12−(12+d)Γ(12−d)]kλ(k(2d−1)+1)(k(2d−1)+2)t2kd−k+2, which is finite, and, consequently, from (4) and (6) we get ∫Rkht(y1,…,yk)2dy1…dyk<2λ[Γ(d)Γ(12−d)π22d]k(k(2d−1)+1)(k(2d−1)+2)t2kd−k+2 for 12−12k<d<12 . Next, suppose d>12 \frac{1}{2}$]]> (equivalently H>1 1$]]>). In this case, ∫0tds∫0λ(t−s)[zd−12K12−d(z)]kdz=∫0tds∫0λ(t−s)[zd−12Kd−12(z)]kdz≤∫0tds∫0λ(t−s)[2d−32Γ(d−12)]kdz≤λ[2d−32Γ(d−12)]k2t2, where we applied the fact that Kν(z)=K−ν(z) and kν(z)<z−ν2ν−1Γ(ν) for z>0 0$]]>. Hence, from (4) and (7) it follows that ∫Rkht(y1,…,yk)2dy1…dyk<[Γ(d)Γ(d−12)2πλ2d−1]kλt2 for d>12 \frac{1}{2}$]]>. Finally, let d=12 (equivalently, H=1 ). Consider ∫0λ(t−s)(K0(z))kdz=∫0η(K0(z))kdz+∫ηλ(t−s)(K0(z))kdz:=I1+I2. Since K0(z)∼−log(z) as z→0 (see [1, Eq. 9.6.8], we have I1=∫0η(K0(z)log(z)log(z))kdz≤(1+ϵ)k∫0η(−log(z))kdz=(1+ϵ)k∫−log(η)+∞wke−wdz≤(1+ϵ)k∫0+∞wke−wdz=(1+ϵ)kΓ(k+1). Now, we find an upper bound for I2 . It can be shown that Kν(x)Kν(y)>ey−x {e}^{y-x}$]]> for 0<x<y and an arbitrary real number ν (see [4]). Therefore, I2=∫ηλ(t−s)(K0(z))kdz<∫ηλ(t−s)(K0(η)eη−z)kdz[K0(η)eη]k(λ(t−s)−η). From (8), (9), and (10) we can see that ∫0λ(t−s)(K0(z))kdz<(1+ϵ)kΓ(k+1)+[K0(η)eη]k(λ(t−s)−η) and hence ∫0tds∫0λ(t−s)[K12−d(z)]kdz<((1+ϵ)kΓ(k+1))t+[K0(η)eη]k(λt22−ηt) for ϵ>0 0$]]>, and this shows that ∫Rkht(y1,…,yk)2dy1…dyk<2[Γ(d)πλ2d−12d−12]k[((1+ϵ)kΓ(k+1))t+[K0(η)eη]k(λt22−ηt)] for d=12 ( H=1 ), which completes the proof.  □

The next result shows that although a tempered Hermite process is not a self-similar process, it does have a nice scaling property. Here the symbol ≜ indicates the equivalence of finite-dimensional distributions.

The process ZH,λk given by (2) has stationary increments such that {ZH,λk(ct)}t∈R≜{cHZH,cλk(t)}t∈R for any scale factor c>0 0$]]>.

Since B(dy) has the control measure m(dy)=σ2dy , the random measure B(cdy) has the control measure c1/2σ2dy . Given tj , j=1,…,n , by the change of variables s=cs′ and yi=cyi′ for i=1…,k we have ZH,λk(ctj)=∫Rk∫0ctj( ∏i=1k(s−yi)+−(12+1−Hk)e−λ(s−yi)+)dsB(dy1)…B(dyk)=∫Rk∫0tj( ∏i=1k(cs′−cyi′)+−(12+1−Hk)e−λ(cs′−cyi′)+)cds′B(cdy1′)…B(cdyk′)≜cH∫R∫0tj((s′−y′)+−(12+1−Hk)e−λc(s′−y′)+)ds′B(dy1′)…B(dyk′)=cHZH,cλk(tj), so that (12) holds. Suppose now that sj<tj and change the variables x=x′+sj , yi=sj+yi′ (for j=1,…,n ) to get (ZH,λk(tj)−ZH,λk(sj))=∫Rk∫sjtj( ∏i=1k(x−yi)+−(12+1−Hk)e−λ(x−yi)+)dxB(dy1)…B(dyk)=∫Rk∫0tj−sj( ∏i=1k(x′+sj−yi)+−(12+1−Hk)e−λ(x′+sj−yi)+)dx′B(dy1)…B(dyk)≜∫Rk∫0tj−sj( ∏i=1k(xi′−yi′)+−(12+1−Hk)e−λ(xi′−yi′)+)dx′B(dy1′)…B(dyk′)=ZH,λk(tj−sj), which shows that a tempered Hermite process of order k has stationary increments.  □

As a consequence of Lemma 1 and Proposition 1, we get the following:

The stochastic process ZH,λk(t) has a continuous version.

According to the proof of Lemma 1, E|ZH,λk(t)−ZH,λk(s)|2≤c1|t−s|2H12<H<1,c2|t−s|2H>1, 1,\end{array}\right.\]]]> where c1 and c2 are some constants. Kolmogorov’s continuity criterion states that a stochastic process X(t) has a continuous version if there exist some positive constants p, β, and c such that E|X(t)−X(s)|p≤c|t−s|1+β. Apply (14) for tempered Hermite process ZH,λk(t) by taking p=1 , β=min{1,2H−1} , and c=min{c1,c2} to get the desired result.  □

We next compute the covariance function of ZH,λk(t) . Unlike Hermite processes, the covariance function of a tempered Hermite process is different for different k≥1 .

The process ZH,λk given by (2) has the covariance function R(t,s)=2[Γ(d)π(2λ)d−12]k∫0t∫0s[|u−v|d−12K12−d(λ|u−v|)]kdvdu for λ>0 0$]]> and d>12−12k \frac{1}{2}-\frac{1}{2k}$]]> (equivalently, H>12 \frac{1}{2}$]]>).

By applying the Fubini theorem and the isometry of multiple Wiener–Itô integrals we have R(t,s)=2∫Rk(∫0t∫0s ∏i=1k(u−yi)+d−1(v−yi)+d−1×e−λ(u−yi)+e−λ(v−yi)+dvdu)dy1…dyk=2∫0t∫0s∫Rk[ ∏i=1k(u−yi)+d−1(v−yi)+d−1×e−λ(u−yi)+e−λ(v−yi)+dy1…dyk]dvdu=2∫0t∫0s[∫R(u−y)+d−1(v−y)+d−1e−λ(u−y)+e−λ(v−y)+dy]kdvdu=2∫0t∫0s[∫−∞min(u,v)(u−y)d−1(v−y)d−1e−λ(u−y)e−λ(v−y)dy]kdvdu=2∫0t∫0s[∫0+∞wd−1(|u−v|+w)d−1e−λwe−λ(|u−v|+w)dw]kdvdu=2∫0t∫0se−λk|u−v||u−v|k(2d−1)×[∫0+∞x−(12+1−Hk)(x+1)−(12+1−Hk)e−2λ|u−v|xdx]kdvdu=2∫0t∫0se−λk|u−v||u−v|k(2d−1)×[Γ(d)π(12λ|u−v|)d−12eλ|u−v|K12−d(λ|u−v|)]kdvdu=2[Γ(d)π(2λ)d−12]k∫0t∫0s[|u−v|d−12K12−d(λ|u−v|)]kdvdu for any H>12 \frac{1}{2}$]]> and λ>0 0$]]>, and hence we get the desired result.  □

Let Bˆ1 and Bˆ2 be independent Gaussian random measures with Bˆ1(A)=Bˆ1(−A) , Bˆ2(A)=−Bˆ2(−A) , and E[(Bˆi(A))2]=m(A)/2 , where m(dx)=σ2dx , and define the complex-valued Gaussian random measure Bˆ=Bˆ1+iBˆ2 .

Let H>12 \frac{1}{2}$]]> and λ>0 0$]]>. The process ZH,λk given by (2) has the spectral domain representation ZH,λk(t)=CH,k∫Rk″eit(ω1+⋯+ωk)−1i(ω1+⋯+ωk)× ∏j=1k(λ+iωj)−(12−1−Hk)Bˆ(dω1)…Bˆ(dωk), where Bˆ(·) is a complex-valued Gaussian random measure, and CH,k=(Γ(12−1−Hk)2π)k is a constant depending on H and k. The double prime ″ on the integral indicates that one does not integrate on the diagonals ωi=ωj , i≠j .

We first observe that ht(y1,…,yk)=∫0t ∏j=1k(s−yj)+d−1e−λ(s−yj)+ds has the Fourier transform htˆ(ω1,…,ωk)=1(2π)k2∫Rkei∑j=1kωjyj∫0t ∏j=1k(s−yj)+d−1e−λ(s−yj)+dsdy1…dyk=1(2π)k2∫Rk∫0tei∑j=1kωj(s−uj) ∏j=1k(uj)+d−1e−λ(uj)+dsdu1…duk=1(2π)k2∫0t∫Rkeis∑j=1kωj ∏j=1k(uj)+d−1e−(λ+iωj)ujdu1…dukds=[Γ(d)2π]keit(ω1+⋯+ωk)−1i(ω1+⋯+ωk) ∏j=1k(λ+iωj)−d=[Γ(12−1−Hk)2π]keit(ω1+⋯+ωk)−1i(ω1+⋯+ωk) ∏j=1k(λ+iωj)−(12−1−Hk), using the well-known formula for the characteristic function of the gamma density. Then (2), together with Proposition 9.3.1 in [21], implies that ZH,λk(t)=∫Rk′ht(y1,…,yk)B(dy1)…B(dyk)≜∫Rk″htˆ(ω1,…,ωk)Bˆ(dω1)…Bˆ(dωk)=CH,k∫Rk″eit(ω1+⋯+ωk)−1i(ω1+⋯+ωk) ∏j=1k(λ+iωj)−(12−1−Hk)Bˆ(dω1)…Bˆ(dωk), which is equivalent to (15).  □

 Limit theorem

In this section, we show that the process ZH,λk(t) is the weak convergence limit of a certain discrete chaos process. Our approach follows the seminal work of Bai and Taqqu [3]. When k=1 and λ>0 0$]]>, the discrete process Yλ,k(n) , (18), is a time series that is useful to model turbulence [20, 24]. When k=1 and λ=0 , Davydov [7] (see also Giraitis et al. [12, p. 276] and Whitt [27, Theorem 4.6.1]) established the corresponding invariance principle for Yλ,k(n) , where the limit involves a fractional Brownian motion. When k>1 1$]]> and λ=0 , Taqqu [26] showed that the weak convergence limit of Yλ,k(n) is the Hermite process (1).

The following proposition gives a powerful tool for proving the result of this section.

Let Qk(gN):= ∑(j1,…,jk)∈Zk′gN(j1,…,jk)εj1…εjk for N=1,2,… , where gN∈L2(Zk) for k≥1 , and {εn} is an i.i.d. sequence with mean zero and variance 1. Assume that, for some f∈L2(Rk) , ∫Rk|g˜N(u1,…,uk)−f(u1,…,uk)|2du1…duk→0,as N→∞, where g˜N(u1,…,uk):=Nk2gN([u1N]+c1,…,[ukN]+ck),(c1,…,ck)∈Zk. Then Qk(gN)⟶f.d.d.∫Rkf(u1,…,uk)B(du1)…B(duk) as N→∞ .

See Proposition 4.1 in [3] and also Corollary 4.7.1 in [12].  □

Define the discrete chaos process Yλ,k(n):= ∑(i1,i2,…,ik)∈Zk′Cλ(i1,i2,…,ik)εn−i1…εn−ik, where the prime ′ indicates exclusion of the diagonals ip=iq , p≠q , {εn} is as before, and Cλ(i1,i2,…,ik)= ∏j=1k(ij)+d−1e−λ(ij)+ for d∈(12−12k,∞) and λ>0 0$]]>. Now, consider SNλ(t)= ∑n=1[Nt]Yλ,k(n),0≤t≤1.

Let Yλ,k(n) be the discrete chaos process given by (18). Then 1NHSNλN(t)⇒ZH,λk(t), where ⇒ means weak convergence in the Skorokhod space D[0,1] with uniform metric, ZH,λk(t) is the tempered Hermite process in (2), and H=1+kd−k2 .

The Lamperti’s theorem [15] states that if 1d(N) ∑k=1[Nt]Yk⟶f.d.d.Z(t) and d(N)→∞ as N→∞ , where {Yk} is stationary, then {Z(t)}t≥0 is self-similar with stationary increments ( ⟶f.d.d. means the convergence of finite-dimensional distributions). In our case, since the stationary processes {YkλN} depend on N through the parameter λ, the limit process {ZH,λk(t)}t≥0 need not be a self-similar process. Therefore, the result of Theorem 1 does not contradict the Lamperti theorem.

First, we show that 1NHSNλN(t)=1NH ∑n=1[Nt]YλN,k(n)=∑(i1,…,ik)∈Zk1NH ∑n=1[Nt]CλN(n−i1,…,n−ik)εi1…εik=Qk(ht,N)⟶f.d.d.ZH,λk(t)asN→∞, where ht,N(i1,…,ik):=1NH ∑n=1[Nt]CλN(n−i1,…,n−ik), and Qk(·) is defined by (17). In order to show (21), we just need to check that ‖h˜t,N(y1,…,yk)−ht(y1,…,yk)‖L2(Rk)→0 as N→∞ , where h˜t,N(y1,…,yk):=Nk2ht,N([Ny1]+1,…,[Nyk]+1)=Nk2NH ∑n=1[Nt]CλN(n−[Ny1]−1,…,n−[Nyk]−1), and ht(y1,…,yk) is given by (3). Write h˜t,N(y1,…,yk)=Nk2NH ∑n=1[Nt]CλN(n−[Ny1]−1,…,n−[Nyk]−1)=1N1+kd−k ∑n=1[Nt] ∏i=1k(n−[Nyi]−1)+d−1e−λN(n−[Nyi]−1)+=1N ∑n=1[Nt] ∏i=1k(n−[Nyi]−1N)+d−1e−λ(n−[Nyi]−1N)+=∫0t ∏i=1k([Ns]−[Nyi]N)+d−1e−λ([Ns]−[Nyi]N)+ds. Let d=1 . In this case, ([Ns]−[Ny]N)+d−1e−λ([Ns]−[Ny]N)+=e−λ([Ns]−[Ny]N)+≤e−λ(s−y)+eλN for all N≥1 , and hence | ∏i=1ke−λ([Ns]−[Nyi]N)+|≤eλk ∏i=1ke−λ(s−yi)+=:g1(s−y1,…,s−yk). Next, consider 0<d<1 . Since [Ns]−[Ny]>Ns−Ny−1 Ns-Ny-1$]]>, we get ([Ns]−[Ny]N)+d−1<(Ns−Ny−1N)+d−1≤(s−y−1)+d−1 for all N≥1 , and hence | ∏i=1k([Ns]−[Nyi]N)+d−1e−λ([Ns]−[Nyi]N)+|< ∏i=1k(s−yi−1)+d−1e−λ(s−yi−1)+=:g2(s−y1,…,s−yk). Finally, suppose that d>1 1$]]>. Since [Ns]−[Ny]<Ns−Ny+1 , we get ([Ns]−[Ny]N)+d−1<(Ns−Ny+1N)+d−1≤(s−y+1)+d−1 for all N≥1 , and hence | ∏i=1k([Ns]−[Nyi]N)+d−1e−λ([Ns]−[Nyi]N)+|< ∏i=1k(s−yi+1)+d−1e−λ(s−yi−1)+=:g3(s−y1,…,s−yk). By the similar argument of Lemma 1, we can verify that ∫Rk′(∫0tgi(s−y1,…,s−yk)ds)2dy1,…,dyk<∞ for i=1,2,3 . On the other hand, since Cλ(i1,…,ik) is continuous a.e., Cλ([Ns]−[Ny1]N,…,[Ns]−[Nyk]N) converges a.e. to Cλ(s−y1,…,s−yk) as N→∞ . Now apply the dominated convergence theorem to get the desired result (22).

In order to show the tightness, we need to verify that E|N−H(SNλN(t)−SNλN(s))|2γ≤C|Fn(t)−Fn(s)|2α,0≤s<t≤1, where γ>0 0$]]> and α>12 \frac{1}{2}$]]> (here {Fn}n≥1 is a sequence of nondecreasing continuous functions on [0,1] that are uniformly bounded and satisfy limδ→0lim supn→∞ωδ(Fn)=0, where ωδ(F):=sup|t−s|<δ|F(t)−F(s)| for δ>0 0$]]>). See Lemma 4.4.1 in [12] for more details. Observe that SNλN(t)= ∑n=1[Nt]YλN,k(n)= ∑(i1,…,ik)∈Z′ ∑n=1[Nt]CλN,k(n−i1,…,n−ik)εi1…εik= ∑(i1,…,ik)∈Z′ ∑n=1[Nt] ∏j=1k(n−ij)+d−1e−λN(n−ij)+εi1…εik= ∑(i1,…,ik)∈Z′Nkd−k+1[1N ∑n=1[Nt] ∏j=1k(n−ijN)+d−1e−λN(n−ij)+]εi1…εik= ∑(i1,…,ik)∈Z′Nkd−k+1[∫0t ∏j=1k([Ny]+1−ijN)+d−1×e−λN([Ny]+1−ij)+dy]εi1…εik=N ∑(i1,…,ik)∈Z′[∫0t ∏j=1k([Ny]+1−ij)+d−1e−λN([Ny]+1−ij)+dy]εi1…εik. Therefore, we get E|N−H(SNλN(t)−SNλN(s))|2=N2−2H×E| ∑(i1,…,ik)∈Z′[∫0t−s ∏j=1k([Ny]+1−ij)+d−1e−λN([Ny]+1−ij)+dy]εi1…εik|2≤k!N2−2H ∑(i1,…,ik)∈Z′[∫0t−s ∏j=1k([Ny]+1−ij)+d−1e−λN([Ny]+1−ij)+dy]2=k!N2−2H+k×∫Rk′[∫0t−s ∏j=1k([Ny]+1−[Nxj])+d−1e−λN([Ny]+1−[Nxj])+dy]2dx1…dxk. Now, we consider two different cases corresponding with the range of d. First, assume that 12−12k<d≤1 (equivalently, 12<H≤1+k2 ): Since [Ny]−[Nxj]+1>Ny−Nxj Ny-Nx_{j}$]]>, we can write E|N−H(SNλN(t)−SNλN(s))|2≤k!N2−2H+k×∫Rk′[∫0t−s ∏j=1k([Ny]+1−[Nxj])+d−1e−λN([Ny]+1−[Nxj])+dy]2dx1…dxk≤k!N2−2H+2kd−k∫Rk′[∫0t−s ∏j=1k(y−xj)+d−1e−λ(y−xj)+dy]2dx1…dxk=k!∫Rk′[∫0t−s ∏j=1k(y−xj)+d−1e−λ(y−xj)+dy]2dx1…dxk. Now, let d>1 1$]]>. Since Ny−Nxj<[Ny]−[Nxj]+1<Ny−Nxj+N , we have E|N−H(SNλN(t)−SNλN(s))|2≤k!N2−2H+k×∫Rk′[∫0t−s ∏j=1k([Ny]+1−[Nxj])+d−1e−λN([Ny]+1−[Nxj])+dy]2dx1…dxk≤k!∫Rk′[∫0t−s ∏j=1k(y−xj+1)+d−1e−λ(y−xj)+dy]2dx1…dxk=k!∫Rk′[∫0t−s ∏j=1k(y−xj+1)d−1e−λ(y−xj+1)eλ1{y>xj}dy]2dx1…dxk=e2λkk!∫Rk′[∫0t−s ∏j=1k(y−zj)d−1e−λ(y−zj)1{y>zj+1}dy]2dz1…dzk≤e2λkk!∫Rk′[∫0t−s ∏j=1k(y−zj)+d−1e−λ(y−zj)+dy]2dz1…dzk. x_{j}\}}\hspace{0.1667em}dy\Bigg]}^{2}\hspace{0.1667em}dx_{1}\dots dx_{k}\\{} & \displaystyle \hspace{1em}={e}^{2\lambda k}k!{\int _{{\mathbb{R}}^{k}}^{{^{\prime }}}}{\Bigg[{\int _{0}^{t-s}}\prod \limits_{j=1}^{k}{(y-z_{j})}^{d-1}{e}^{-\lambda (y-z_{j})}\mathbf{1}_{\{y>z_{j}+1\}}\hspace{0.1667em}dy\Bigg]}^{2}\hspace{0.1667em}dz_{1}\dots dz_{k}\\{} & \displaystyle \hspace{1em}\le {e}^{2\lambda k}k!{\int _{{\mathbb{R}}^{k}}^{{^{\prime }}}}{\Bigg[{\int _{0}^{t-s}}\prod \limits_{j=1}^{k}{(y-z_{j})_{+}^{d-1}}{e}^{-\lambda (y-z_{j})_{+}}\hspace{0.1667em}dy\Bigg]}^{2}\hspace{0.1667em}dz_{1}\dots dz_{k}.\end{array}\]]]> Therefore, E|N−H(SNλN(t)−SNλN(s))|2≤e2λkk!∫Rk′[∫0t−s ∏j=1k(y−zj)+d−1e−λ(y−zj)+dy]2dz1…dzk for any d>12−12k \frac{1}{2}-\frac{1}{2k}$]]> (equivalently, H>12 \frac{1}{2}$]]>). According to the proof of Lemma 1, e2λkk!∫Rk′[∫0t−s ∏j=1k(y−zj)+d−1e−λ(y−zj)+dy]2dz1…dzk≤C|t−s|2H for 12<H<1 and e2λkk!∫Rk′[∫0t−s ∏j=1k(y−zj)+d−1e−λ(y−zj)+dy]2dz1…dzk≤C|t−s|2 for H>1 1$]]>. Now, it remains to apply (23) by selecting γ=1 , α=min{H,1} , and Fn(t)=t to get the desired result.  □

 Acknowledgments

The author would like to thank Hira Koul and Mark Meerschaert from Michigan State University for fruitful discussions.

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