# A Weak Convergence to Hermite Process by Martingale Differences

Authors: Xichao Sun; Ronglong Cheng
Journal: Advances in Mathematical Physics
 (2014)

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## Abstract

We consider the weak convergence to general Hermite processZH,k of order k with index H. By applying martingale differences we construct a sequence {ZH,kn,n=1,2,…} of multiple Wiener-Itô stochastic integrals such that it converges in distribution to the Hermite process ZH,k.


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## Body

## 1. Introduction

The self-similar stochastic processes, which are exactly of approximately similar to a part of themselves, have been widely studied since their practical applications to internet traffic, hydrology, turbulence, and finance. We refer to the work of Taqqu [1] for a guide on the appearance of the self-similarity in many applications and to the monographs by Embrechts and Maejima [2] and by Samorodnitsky and Taqqu [3] for complete expositions on self-similar processes. In this paper, we consider the so-called Hermite process with index H∈(1/2,1), which is a special class of self-similar processes with long range dependency. This process is given as limits of the so calledNon Central Limit Theorem studied by Dobrushin and Major [4] and Taqqu [5].Consider a stationary centered Gaussian sequence(ξn)n∈(N) with E(ξn2)=1 such that
(1)r(n)≔E(ξ0ξn)=n(2H-2)/kL(n),
where k≥1 are integers, H∈(1/2,1), and L is a slowly varying function at infinity. Denote by Hm(x) the Hermite polynomial of degree m defined by
(2)Hm(x)=(-1)mex2/2dmdxme-x2/2,m=1,2,…
and H0(x)=1. Let the function g:R→R satisfy E(g(ξ0))=0 and E(g(ξ0)2)<∞. Suppose that g has Hermite rank equal to k, that is, if g admits the following expansion in Hermite polynomials:
(3)g(x)=∑j=0∞‍cjHj(x),cj=1j!E(g(ξ0Hj(ξ0))).
The Hermite rank of g is defined by
(4)k=min⁡{j;cj≠0}.
Since
(5)E[g(ξ0)]=0,
we have k≥1. Then, theNon Central Limit Theorem (see [4, 5]) implies that the sequence of stochastic processes of the form
(6)1nH∑j=1[nt]‍g(ξj),n=1,2,…
converges, as n→∞, in the sense of finite dimensional distributions to the process
(7)ZH,k(t)=∫[0,t]k‍Q(t,s1,…,sk)dB(s1)⋯dB(sk),t∈[0,1],
where the integral is a multiple Wiener-Itô stochastic integral with respect to standard Brownian motion B={B(t),t≥0} and
(8)Q(t,s1,…,sk)=c(H,k)(∏j=1k‍sj(1/2)-H0)×∫0t‍uk(H0-(1/2))∏j=1k‍(u-sj)+H0-(3/2)du
with H0=1-((1-H)/k)∈(1-(1/2k),1), s1,…,sk∈[0,t] and a positive normal constant c(H,k) such that E[(ZH,k(1))2]=1.Definition 1 (see [5]).
Letk≥1 be integer and let H∈(1/2,1). The process (ZH,k(t))t≥0 defined by (7) is called the Hermite process of order k, with index H.Clearly, whenk=1 the Hermite process is the fractional Brownian motion with Hurst parameter H∈(1/2,1). When k=2 the Hermite process is called the Rosenblatt process (see [6]). It is important to note that the Hermite process is not Gaussian for k≥2. The simplest Hermite process is the fractional Brownian motion. The Rosenblatt process is the simplest non-Gaussian Hermite process. More work for the Hermite process and related processes can be found in Bardet and Tudor [7], Chronopoulou et al. [8], Li [9, 10], Li and Lim [11], Heydari et al. [12, 13], Maejima and Tudor [14], Pipiras and Taqqu [15], Torres and Tudor [16], Tudor [17], Tudor and Viens [18], and the references therein. The Hermite processes are neither a semimartingale nor a Markov process, and they admit the same properties as Gaussian fractional Brownian motion. We mention the following:(i)
it exhibits long-range dependence in the sense that(9)∑n≥α‍ρn(α)=∞for anyα>0, where
(10)ρn(α)=E[(ZH,k(α+1)-ZH,k(α))(ZH,k(n+1)-ZH,k(n))].(ii)
it isH-selfsimilar in the sense that for any c>0,
(11){ZH,k(ct),t≥0}=d{cHZH,k(t),t≥0},where=d indicates here equality of the finite-dimensional distributions;(iii)
it has stationary increments, that is, the joint distribution of(ZH,k(t+h)-ZH,k(t),h≥0) is independent of every t≥0;(iv)
its covariance function is(12)E[ZH,k(t)ZH,k(s)]=12[t2H+s2H-|t-s|2H];(v)
it is Hölder continuous of orderγ<H. These properties of the Hermite process motivate our interesting to study them.  Among the applications of the Hermite processes in statistics or econometrics, for example, see Hall et al. [19], Leonenko and Woyczynski [20], and Tudor [17]. Besides these more or less practical applications of the Hermite processes, our motivation is also theoretical; it comes from the recent intensive interest to push further the stochastic calculus with respect to more and more general integrator processes. We believe that such class of processes constitutes an interesting and instructive example where the recent developed techniques of the generalized stochastic calculus can find a significant test bench. On the other hand, the fractional Brownian motion has become an object of intense study in recent some years, due to its interesting properties and its applications in various scientific areas including telecommunications, turbulence, image processing, and finance. Some surveys and complete literature for fractional Brownian motion could be found in Alós et al. [21], Biagini et al. [22], Hu [23], Mishura [24], Nieminen [25], Nualart [26], and the references therein. However, in contrast to the extensive studies on fractional Brownian motion, there has been little systematic investigation on the general Hermite process. The main reasons for this, in our opinion, are the complexity of dependence structures and property of non-Gaussian. Therefore, it seems interesting to study the Hermite process. In this short paper we will prove a weak convergence theorem to the Hermite process based on martingale differences. Our main object is to explain and prove the following theorem.Theorem 2.
Let(ξ(n))n≥1 be a sequence of square integrable martingale differences satisfying the following conditions:
(13)lim⁡n→∞(ξi(n))21/n=1a.s.
for every i≥1 and
(14)max⁡1≤i≤n|ξi(n)|≤Cna.s.
for some C≥1. Define the two sequences {Wn} and {Zn} of processes as follows:
(15)Wtn≔∑i=1[nt]‍ξi(n),0≤t≤1,Ztn≔∫Rk‍Qn(t,s1,…,sk)dWs1n⋯dWskn,0≤t≤1,
where Qn(t,s1,…,sk) is given by
(16)Qn(t,s1,…,sk)=c(H,k)nk×∫((s1-1)/n)(s1/n)‍⋯∫((sk-1)/n)(sk/n)‍Q([nt]n,r1,…,rk)vvvvvvvvvvvvvvvvvvvwwwwwwv×dr1⋯drk,
with s1,…,sk∈[1/n,1]. Then the process Zn converges in distribution to the Hermite process ZH,k as n tends to infinity.Remark 3.
One eliminates the diagonal “s1≠⋯≠sk” because the Hermite process is defined as a multiple Wiener-Itô integral as a consequence it has zero mean.This paper is organized as follows. In Section2 we present some preliminaries for the Hermite process. In Section 3 we will give the proof of Theorem 2.


## 2. Hermite Processes

In this section, we briefly recall some basic properties of the Hermite process. We refer to Maejima and Tudor [14], Peccati and Taqqu [27], and Pipiras and Taqqu [15] for some complete descriptions of the Hermite processes. For simplicity we let C stand for a positive constant depending only on subscripts and its value may be different in different appearances. Throughout this paper we assume that 1/2<H<1, and let ZH,k={ZH,k(t)}t∈R be a Hermite process of order k with index H∈(1/2,1). As we pointed out before, we shall use Hermite processes to illustrate the preceding discussion. The Hermite processes {ZH,k(t),t≥0} are self-similar processes with parameter H∈(1/2,1), that is, for any c>0,
(17){ZH,k(ct),t≥0}=d{cHZH,k(t),t≥0},
where =d indicates here equality of the finite-dimensional distributions. They have also stationary increments and are represented by multiple integrals of order k and hence are non-Gaussian when k≥2. The simplest Hermite process is fractional Brownian motion. The Rosenblatt process is the simplest non-Gaussian Hermite process. The Hermite processes are neither a semimartingale nor a Markov process. They have the four representations, which are as follows (see [15, 27]):(i)
time domain representation: fort∈R(18)ZH,k(t)=∫Rk‍f(t,s1,…,sk)dB(s1)⋯dB(sk),where the integral is multiple Wiener-Itô integral of orderk with respect to the standard Brownian motion B((t))t∈R and the kernel f is defined as
(19)f(t,s1,…,sk)≔c(H,k)∫0t‍(∏j=1k‍(u-sj)+H0-(3/2))du,withH0=1-((1-H)/k)∈(1-(1/2k),1), s1,…,sk∈R, x+=max⁡{x,0}, and
(20)c(H,k)=(H(2H-1)[Γ((1/2)+((1-H)/k))]kk![Γ((1/2)-((1-H)/k))Γ(2(1-H)/k)]k)1/2;(ii)
positive half-axis representation: fort≥0(21)ZH,k(t)=∫[0,∞)k‍g(t,s1,…,sk)dB(s1)⋯dB(sk),where the kernelg is defined as
(22)g(t,s1,…,sk)≔c(H,k)∫0t(∏j=1ksj(1/2)-H0(1-usj)+H0-(3/2))duwiths1,…sk∈R+;(iii)
spectral domain representation: fort∈R(23)ZH,k(t)=c^(H,k)∫Rd‍ei(λ1+⋯+λk)t-1i(λ1+⋯+λk)∏j=1k‍|λj|(1/2)-H0vvvvvvvvvvvvvvv×dW^(λ1)⋯dW^(λk),whereW^ is a suitable complex-valued Brownian motion on R and
(24)c^(H,k)=(×sin((12-1-Hk)π)]kkH(2H-1)ffff×(×sin((12-1-Hk)π)]k)1/2k![2Γ(2(1-H)k)ffffffffffff×sin((12-1-Hk)π)]k)-1)1/2;(iv)
finite time interval representation: fort≥0,
(25)ZH,k(t)=∫[0,t]kQ(t,s1,…,sk)dB(s1)⋯dB(sk),t∈[0,1],where the integral is a multiple Wiener-Itô stochastic integral with respect to standard Brownian motionB={B(t),t≥0} and
(26)Q(t,s1,…,sk)=c(H,k)(∏j=1k‍sj(1/2)-H0)×∫0tuk(H0-(1/2))∏j=1k‍(u-sj)+H0-(3/2)duwiths1,…sk∈[0,t].


## 3. Proof of Theorem2

For simplicity we letC stand for a positive constant depending only on subscripts and its value may be different in different appearances. Denote by D=D(0,1) the Skorohod space of right continuous functions on the interval [0,1] with left-hand limits and equip D with the metric
(27)d(u,v)≔inf⁡{andsup⁡t|u(t)-v(λ(t))|≤ɛɛ>0:∃λ∈Λsuchthat∥λ∥≤ɛvvvvvvvvandsup⁡t|u(t)-v(λ(t))|≤ɛ},
where ∥λ∥≔sup⁡s≠t|log⁡(λ(t)-λ(s))/(t-s)| and
(28)Λ≔{λ:[0,1]⟶[0,1],vvvvvvastrictlyincreasingandcontinuousmapping}.
Under this metric, D is a separable and complete metric space. Let X,Xn:Ω→D be random variables and
(29)TX≔{t∈(0,1):P(Xt≠Xt-)=0}∪{0,1}.
Recall that Xn converges in distribution to X if, for every bounded and continuous ϕ:D→R,
(30)E(ϕ(Xn))⟶E(ϕ(X)),
as n tends to infinity. In order to prove Theorem 2 we need the three basic results. The first one is in Billingsley [28] which basically states that weak convergence in D can be proved by using the convergence of finite-dimensional distributions and the tightness of a sequence.Theorem 4.
Suppose thatX,Xn:Ω→D are random variables and
(31)(Xt1n,…,Xtkn)→d(Xt1,…,Xtk)
holds whenever t1,…,tk∈TX, where →d means standard convergence in distribution. Assume that P{X1≠X1-}=0 and that
(32)E{|Xtn-Xt1n|C|Xt2n-Xtn|C}≤[F(t2)-F(t1)]2α,
for t1≤t≤t2 and n≥1, where C>0,α>(1/2) and F is a nondecreasing, continuous function on [0,1]. Then
(33)Xn→dX
in the sense of (30).Secondly, we need the Lindeberg condition (see [29]).Theorem 5.
Lett∈(0,1], σt2≥0 and let (ξ(n))n≥1 be a sequence of square integrable martingale differences satisfying the Lindeberg condition:
(34)∑i=1[nt]‍E[(ξi(n))2I{|ξi(n)|>ɛ}∣Fi-1n]→P0
for ɛ>0. Then ∑i=1[nt]‍(ξi(n))2→Pσt2 implies that the convergence
(35)Wtn≔∑i=1[nt]‍ξi(n)→dN~N(0,σt2)
holds, as n tends to infinity.Thirdly, we need the following auxiliary lemma.Lemma 6.
LetQ be as in (8) and Let (ξ(n))n≥1 satisfy conditions (13) and (14). Then, as n tends to infinity
(36)∑i1,…,ik=1,i1≠⋯≠ikn‍n2k∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q([ntu]n,s1,…,sk)ds1⋯dsk×∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q([ntv]n,s1,…,sk)vvvvvvvvvvvvvvvvvvvv×ds1⋯dsk(ξi1(n)⋯ξik(n))2⟶∫01‍⋯∫01‍Q(tu,s1,…,sk)Q(tv,s1,…,sk)ds1⋯dsk
holds almost surely, for tu,tv∈[0,1].Proof.
Let us taketu=tv and prove the first the case, that is,
(37)∑i1,…,ik=1,i1≠⋯≠ikn‍n2k(∫((i1-1)/n)(i1/n)‍vvvvvvvvvvvvvvvivv⋯∫((ik-1)/n)(ik/n)‍Q(tu,s1,…,sk)ds1⋯dsk)2×(ξi1(n)⋯ξik(n))2⟶∫01‍⋯∫01‍Q2(tu,s1,…,sk)ds1⋯dsk,
holds a.s. as n→∞. For every t1,…,tk∈[0,1], define
(38)gn(t1,…,tk)≔nk×∑i1,…,ik=1,i1≠⋯≠ikn‍I(((i1-1)/n),(i1/n)](t1)⋯I(((ik-1)/n),(ik/n)](tk)×∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tu,s1,…,sk)ds1⋯dsk×(ξi1(n)1/n)⋯(ξik(n)1/n).
Then one has
(39)gn2(t1,…,tk)=(nk∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tu,s1,…,sk)ds1⋯dsk)2×(ξi1(n)1/n)2⋯(ξik(n)1/n)2,
for tj∈((ij-1)/n,ij/n],j=1,…,k and
(40)∫01‍⋯∫01‍gn2(s1,…,sk)ds1⋯dsk=∑i1,…,ik=1,i1≠⋯≠ikn‍n2k×(∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tu,s1,…,sk)ds1⋯dsk)2×(ξi1(n)⋯ξik(n))2.
Firstly, we prove that gn2(t1,…,tk) is a.s. uniformly integrable. In fact, together with Hölder’s inequality, condition (14), and Fubini theorem, we get
(41)∫01‍⋯∫01‍gn2(s1,…,sk)ds1⋯dsk=∑i1,…,ik=1,i1≠⋯≠ikn‍n2k(ξi1(n)⋯ξik(n))2×(∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tu,s1,…,sk)ds1⋯dsk)2≤C∑i1,…,ik=1,i1≠⋯≠ikn‍∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q2(tu,s1,…,sk)vvvvvvvvvvvvvvvvvvvvvvvvvwwwvvvvvv×ds1⋯dsk≤C∫[0,1]k‍(∏j=1k‍sj1-2H0∫0tu‍∫0tu‍xk(H0-(1/2))yk(H0-(1/2))vvvvvvvvvvvv×∏j=1k‍(x-sj)+H0-(3/2)(y-sj)+H0-(3/2)dxdy)×ds1⋯dsk=C∫0tu‍∫0tu‍xk(H0-(1/2))yk(H0-(1/2))×[∫[0,1]k‍∏j=1k‍sj1-2H0(x-sj)+H0-(3/2)(y-sj)+H0-(3/2)vvvvvvvvv×ds1⋯dsk∏j=1k‍]dxdy=C∫0tu‍∫0tu‍xk(H0-(1/2))yk(H0-(1/2))×[∫01‍s1-2H0(x-s)+H0-(3/2)(y-s)+H0-(3/2)ds]kdxdy≤C∫0tu‍∫0tu‍|x-y|(2H0-2)kdxdy=C∫0tu‍∫0tu‍|x-y|2H-2dxdy≤C∫01‍∫01‍|x-y|2H-2dxdy<∞,
since 1/2<H<1, where we have used the fact
(42)∫0x∧y‍s1-2H0(x-s)+H0-(3/2)(y-s)+H0-(3/2)ds=β(2-2H0,H0-(1/2))(xy)H0-(1/2)|x-y|2H0-2.
In addition, for tj∈((ij-1)/n,ij/n],j=1,…,k,
(43)gn2(t1,…tk)=(nk∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tu,s1,…,sk)ds1⋯dsk)2×(ξi1(n)1/n)2⋯(ξik(n)1/n)2⟶Q2(tu,s1,…,sk)
holds a.s., as n tends to infinity, because
(44)nk∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tu,s1,…,sk)ds1⋯dsk⟶Q(tu,s1,…,sk),(ξij(n)1/n)2⟶1,j=1,2,…,k
holds a.s., as n tends to infinity. Thus
(45)∫01‍⋯∫01‍gn2(s1,…,sk)ds1⋯dsk⟶∫01‍⋯∫01‍Q2(tu,s1,…,sk)ds1⋯dsk
holds a.s., as n tends to infinity, and (37) follows.
Next we will prove the original claim. Denote thattun:=[ntu]/n and tvn:=[ntv]/n. From the previous discussion we know that when n→∞,
(46)∑i1,…,ik=1,i1≠⋯≠ikn‍n2k∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tu,s1,…,sk)ds1⋯dsk×∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tv,s1,…,sk)ds1⋯dsk×(ξi1(n)⋯ξik(n))2⟶∫01‍⋯∫01‍Q(tu,s1,…,sk)vvvvvvvvvvvv×Q(tv,s1,…,sk)ds1⋯dsk,a.s..
It turns into show that the difference
(47)∑i1,…,ik=1,i1≠⋯≠ikn‍n2k×(∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tu,s1,…,sk)ds1⋯dskvvvvv×∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tv,s1,…,sk)ds1⋯dskvvvvv-∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tun,s1,…,sk)ds1⋯dskvvvvv×∫((i1-1)/n)(i1/n)‍vvvvv⋯∫((ik-1)/n)(ik/n)‍Q(tvn,s1,…,sk)ds1⋯dsk)vvvvv×(ξi1(n)⋯ξik(n))2
tends to zero a.s. as n tends to infinity.
SinceQ is increasing with respect to t and tu≥tun,tv≥tvn, we can get
(48)0≤∑i1,…,ik=1,i1≠⋯≠iknn2k‍×(∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tu,s1,…,sk)ds1⋯dskvvv×∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tv,s1,…,sk)ds1⋯dskvvv-∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tun,s1,…,sk)ds1⋯dskvvv×∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tvn,s1,…,sk)ds1⋯dsk)×(ξi1(n)⋯ξik(n))2≤∑i1,…,ik=1,i1≠⋯≠ikn‍n2k∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tu,s1,…,sk)ds1⋯dsk(ξi1(n)⋯ξik(n))2×∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍(Q(tv,s1,…,sk)-Q(tvn,s1,…,sk))vvvvvvv×ds1⋯dsk+∑i1,…,ik=1,i1≠⋯≠ikn‍n2k(ξi1(n)⋯ξik(n))2×∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍|Q(tvn,s1,…,sk)-Q(tv,s1,…,sk)|vvvvvvvvvvv×ds1⋯dsk×∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍|Q(tu,s1,…,sk)-Q(tun,s1,…,sk)|vvvvvvvvvvv×ds1⋯dsk+∑i1,…,ik=1,i1≠⋯≠ikn‍n2k∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(tv,s1,…,sk)ds1⋯dsk(ξi1n⋯ξikn)2×∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍|Q(tu,s1,…,sk)-Q(tun,s1,…,sk)|vvvvvvvv×ds1⋯dsk≡I+II+III.
Notice that
(49)∫0tu‍∫tvntv‍xk(H0-(1/2))yk(H0-(1/2))×[∫01‍s1-2H0(x-s)+H0-(3/2)(y-s)+H0-(3/2)ds]kdxdy≤∫0tu‍∫tvntv‍|x-y|(2H0-2)kdxdy=∫0tu‍∫tvntv‍|x-y|2H-2dxdy≤∫01‍∫01‍|x-y|2H-2I(0,tu](x)I([ntv]/n,tv](y)dxdy,tu≥tun,tv≥tvn. We can get, by using the first part of this proof,
(50)I⟶∫01‍…∫01‍Q(tu,s1,…,sk)vvvvvvvvvvv×(Q(tv,s1,…,sk)-Q(tvn,s1,…,sk))vvvvvvvvvvv×ds1⋯dsk≤C∫[0,1]k‍(∏j=1k‍sj1-2H0∫0tu‍∫tvntv‍xk(H0-(1/2))yk(H0-(1/2))vvvvvvvivv×∏j=1k‍(x-sj)+H0-(3/2)(y-sj)+H0-(3/2)dxdy)vvvvvvv×ds1⋯dsk=C∫0tu‍∫tvntv‍xk(H0-(1/2))yk(H0-(1/2))vvvvvvivvv×[∫[0,1]k‍∏j=1k‍sj1-2H0(x-sj)+H0-(3/2)vvvvvvvvvvvvvvvv×(y-sj)+H0-(3/2)ds1⋯dsk∏j=1k‍]dxdy=C∫0tu‍∫tvntv‍xk(H0-(1/2))yk(H0-(1/2))vvvvvvvvv×[∫01‍s1-2H0(x-s)+H0-(3/2)(y-s)+H0-(3/2)ds]kvvvvvvvvv×dxdy⟶0,
as n tends to infinity because 0≤tv-[ntv]/n≤1/n. Similarly the last two summands II and III also tend to zero and we have that when n→∞,
(51)∑i1,…,ik=1,i1≠…≠ikn‍n2k∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q([ntu]n,s1,…,sk)ds1⋯dsk×∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q([ntv]n,s1,…,sk)ds1⋯dsk×(ξi1(n)⋯ξik(n))2⟶∫01‍⋯∫01‍Q(tu,s1,…,sk)Q(tv,s1,…,sk)ds1⋯dsk
holds a.s., and the lemma follows.We can prove the main result in this paper.Proof of Theorem2.
We prove this theorem by using Theorem4. We first have to prove that the finite-dimensional distributions of Zn converge to those of ZH,k.
For anya1,…,ad∈R and t1,…,td∈[0,1], we want to show that the the linear combination
(52)Yn≔∑u=1d‍auZtun
converges in distribution to a normally distributed random variable with expectation zero and variance
(53)E(∑u=1d‍auZtu)2.
The fact that the expectation is zero is trivial. Let us write Yn as
(54)Yn=∑i1,…,ik=1,i1≠⋯≠ikn‍nk(ξi1(n)⋯ξik(n))2×∑u=1d‍au∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q([nt]n,s1,…,sk)ds1⋯dsk≔∑i1,…,ik=1,i1≠⋯≠ikn‍Yi1,…,ikn.
The Lindeberg condition is satisfied if for anyɛ>0, we have
(55)∑i1,…,ik=1,i1≠⋯≠ikn‍E[(Yi1,…,ikn)2I{|Yi1,…,ikn|>ɛ}∣Fi1-1,…,ik-1n]→p0,
as n tends to infinity. Consider the set
(56){|Yi1,…,ikn|>ɛ}={(Yi1,…,ikn)2>ɛ2}.
We get an upper bound to Yi1,…,ikn by noticing that Q(t,s1,…,sk) is increasing with respect to t and decreasing with respect to s1,…,sk.
Consider the following:(57)(Yi1,…,ikn)2=(∑u=1d‍nk(ξi1(n)⋯ξik(n))2∑u=1d‍au∫((i1-1)/n)(i1/n)‍vvvvvv⋯∫((ik-1)/n)(ik/n)‍Q([nt]n,s1,…,sk)ds1⋯dsk)2≤n2k(ξi1(n)⋯ξik(n))2(∑u=1d‍|au|)2×(∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q(1,s1,…,sk)ds1⋯dsk)2≤nk(ξi1(n)⋯ξik(n))2(∑u=1d‍|au|)2×∫0(1/n)‍⋯∫0(1/n)‍Q2(1,s1,…,sk)ds1⋯dsk=:nk(ξi1(n)⋯ξik(n))2Aδn,
where A≔(∑u=1d‍|au|)2 and δn≔∫01/n‍⋯∫01/n‍Q2(1,s1,…,sk)ds1⋯dsk. Thus we obtain
(58){|Yi1,…,ikn|>ɛ}⊂{nk(ξi1(n)⋯ξik(n))2Aδn>ɛ2}.
Combining this with the Cauchy-Schwartz inequality, we can get that
(59)E[(Yi1,…,ikn)2I{|Yi1,…,ikn|>ɛ}∣Fi1-1,…,ik-1n]=(nk∑u=1d‍au∫((i1-1)/n)(i1/n)‍vvvvv⋯∫((ik-1)/n)(ik/n)‍Q([nt]n,s1,…,sk)ds1⋯dsk)2×E[(ξi1(n)⋯ξik(n))2I{|Yi1,…,ikn|>ɛ}∣Fi1-1,…,ik-1n]≤(nk∑u=1d‍au∫01/n‍⋯∫01/n‍Q(1,s1,…,sk)ds1⋯dsk)2×(C2n)kE[I{|Yi1,…,ikn|>ɛ}∣Fi1-1,…,ik-1n]≤C2kAδnE[I{nk(ξi1(n)⋯ξik(n))2Aδn>ɛ2}∣Fi1-1,…,ik-1n],
a.s., which deduces that the Lindeberg condition is satisfied:
(60)∑i1,…,ik=1,i1≠⋯≠ikn‍E[(Yi1,…,ikn)2I{|Yi1,…,ikn|>ɛ}∣Fi1-1,…,ik-1n]≤∑i1,…,ik=1,i1≠⋯≠ikn‍C2kAδnE[I{nk(ξi1(n)⋯ξik(n))2Aδn>ɛ2}∣Fi1-1,…,ik-1n]≤C2kAδn∑i1,…,ik=1,i1≠⋯≠ikn‍I{C2kAδn>ɛ2}⟶0,
as n tends to infinity, because of δn→0 and I{C2kAδn>ɛ2}=0 for large n. Moreover, we have
(61)∑i1,…,ik=1,i1≠⋯≠ikn‍(Yi1,…,ikn)2=∑u=1,v=1d‍auav∑i1,…ik=1,i1≠⋯≠ikn‍n2k∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q([ntu]n,s1,…,sk)ds1⋯dsk×∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍Q([ntv]n,s1,…,sk)ds1⋯dsk×(ξi1(n)⋯ξik(n))2,
for all n≥1. By Lemma 6 and the fact that
(62)EZtuZtv=∫0min⁡(tu,tv)‍⋯∫0min⁡(tu,tv)‍Q(tu,s1,…,sk)Q(tv,s1,…,sk)vvvvvvvv×ds1⋯dsk,
it follows that the sum ∑i1,…,ik=1,i1≠⋯≠ikn(Yi1,…,ikn)2 converges to
(63)∑u=1,v=1d‍auav∫01‍⋯∫01‍Q(tu,s1,…,sk)Q(tv,s1,…,sk)ds1⋯dsk=E(∑u=1d‍auZtu)2.
So by Theorem 5, the finite-dimensional distributions of Zn converge to those of ZH,k.
Finally, we need to prove the tightness of the sequenceZn in D. Let s,t∈[0,1], s<t. It is easy to prove that
(64)E[ξi1(n)ξj1(n)⋯ξik(n)ξjk(n)]=0
provided il≠jl for some l=1,2,…,k. It follows from the Cauchy-Schwartz inequality that
(65)E(Ztn-Zsn)2=E(∑i1,…,ik=1,i1≠⋯≠ik[nt]‍nk∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍ξi1(n)⋯ξik(n)vvvvvv×[Q([nt]n,s1,…,sk)vvvvvvvvvv-Q([ns]n,s1,…,sk)]ds1⋯dsk∑i1,…,ik=1,i1≠⋯≠ik[nt]‍)2=∑i1,…,ik=1,i1≠⋯≠ik[nt]‍E(ξi1(n)⋯ξik(n))2×(nk∫((i1-1)/n)(i1/n)‍vvvv⋯∫((ik-1)/n)(ik/n)‍[Q([nt]n,s1,…,sk)vvvvvvvvvvvvvvvv-Q([ns]n,s1,…,sk)]ds1⋯dsk∫((i1-1)/n)(i1/n)‍)2≤∑i1,…,ik=1,i1≠⋯≠ik[nt]‍nkE(ξi1(n)⋯ξik(n))2×∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍[Q([nt]n,s1,…,sk)vvvvvvvvvvvvv-Q([ns]n,s1,…,sk)]2ds1⋯dsk≤∑i1,…,ik=1,i1≠⋯≠ik[nt]‍C2k×∫((i1-1)/n)(i1/n)‍⋯∫((ik-1)/n)(ik/n)‍[Q([nt]n,s1,…,sk)vvvvvvvvvvvvv-Q([ns]n,s1,…,sk)]2ds1⋯dsk≤C2k∫01‍⋯∫01‍[Q([nt]n,s1,…,sk)vvvvvvvvvvvvvvvv-Q([ns]n,s1,…,sk)]2ds1⋯dsk≤C2k|[nt]n-[ns]n|2H.
Notice that for any s<t<r,
(66)E|Ztn-Zsn||Zrn-Ztn|≤(E(Ztn-Zsn)2)1/2(E(Zrn-Ztn)2)1/2≤C2k|[nt]n-[ns]n|H|[nr]n-[nt]n|H≤C2k|[nr]n-[ns]n|2H.
We see that
(67)E|Ztn-Zsn||Zrn-Ztn|≤C2k|r-s|2H,
if r-s≥1/n. On the other hand if r-s<1/n, then either s and t or t and r belong to a same subinterval [m/n,(m+1)/n] for some integer m. Thus the left-hand side of (67) is zero. Therefore, (67) holds for all s<t<r and Zn→dZH,k by Theorem 4. This completes the proof.



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*Source: 307819-2014-07-14.xml*