A Weak Convergence to Hermite Process by Martingale Differences

Abstract

We consider the weak convergence to general Hermite process of order with index . By applying martingale differences we construct a sequence of multiple Wiener-Itô stochastic integrals such that it converges in distribution to the Hermite process .

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 , which is a special class of self-similar processes with long range dependency. This process is given as limits of the so called Non Central Limit Theorem studied by Dobrushin and Major [4] and Taqqu [5].

Consider a stationary centered Gaussian sequence with such that where are integers, , and is a slowly varying function at infinity. Denote by the Hermite polynomial of degree defined by and . Let the function satisfy and . Suppose that has Hermite rank equal to , that is, if admits the following expansion in Hermite polynomials: The Hermite rank of is defined by Since we have . Then, the Non Central Limit Theorem (see [4, 5]) implies that the sequence of stochastic processes of the form converges, as , in the sense of finite dimensional distributions to the process where the integral is a multiple Wiener-Itô stochastic integral with respect to standard Brownian motion and with , and a positive normal constant such that .

Definition 1 (see [5]). Let be integer and let . The process defined by (7) is called the Hermite process of order , with index .

Clearly, when the Hermite process is the fractional Brownian motion with Hurst parameter . When the Hermite process is called the Rosenblatt process (see [6]). It is important to note that the Hermite process is not Gaussian for . 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  for any , where (ii)it is -selfsimilar in the sense that for any ,  where indicates here equality of the finite-dimensional distributions;(iii)it has stationary increments, that is, the joint distribution of is independent of every ;(iv)its covariance function is (v)it is Hölder continuous of order . 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 be a sequence of square integrable martingale differences satisfying the following conditions: for every and for some . Define the two sequences and of processes as follows: where is given by with . Then the process converges in distribution to the Hermite process as tends to infinity.

Remark 3. One eliminates the diagonal “” 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 Section 2 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 stand for a positive constant depending only on subscripts and its value may be different in different appearances. Throughout this paper we assume that , and let be a Hermite process of order with index . As we pointed out before, we shall use Hermite processes to illustrate the preceding discussion. The Hermite processes are self-similar processes with parameter , that is, for any , where indicates here equality of the finite-dimensional distributions. They have also stationary increments and are represented by multiple integrals of order and hence are non-Gaussian when . 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: for  where the integral is multiple Wiener-Itô integral of order with respect to the standard Brownian motion and the kernel is defined as  with , , , and (ii)positive half-axis representation: for  where the kernel is defined as  with ;(iii)spectral domain representation: for  where is a suitable complex-valued Brownian motion on and (iv)finite time interval representation: for ,  where the integral is a multiple Wiener-Itô stochastic integral with respect to standard Brownian motion and  with .

3. Proof of Theorem 2

For simplicity we let stand for a positive constant depending only on subscripts and its value may be different in different appearances. Denote by the Skorohod space of right continuous functions on the interval with left-hand limits and equip with the metric where and Under this metric, is a separable and complete metric space. Let be random variables and Recall that converges in distribution to if, for every bounded and continuous , as 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 can be proved by using the convergence of finite-dimensional distributions and the tightness of a sequence.

Theorem 4. Suppose that are random variables and holds whenever , where means standard convergence in distribution. Assume that and that for and , where and F is a nondecreasing, continuous function on . Then in the sense of (30).

Secondly, we need the Lindeberg condition (see [29]).

Theorem 5. Let , and let be a sequence of square integrable martingale differences satisfying the Lindeberg condition: for . Then implies that the convergence holds, as tends to infinity.

Thirdly, we need the following auxiliary lemma.

Lemma 6. Let be as in (8) and Let satisfy conditions (13) and (14). Then, as tends to infinity holds almost surely, for .

Proof. Let us take and prove the first the case, that is, holds a.s. as . For every , define Then one has for and Firstly, we prove that is a.s. uniformly integrable. In fact, together with Hölder’s inequality, condition (14), and Fubini theorem, we get since , where we have used the fact In addition, for , holds a.s., as tends to infinity, because holds a.s., as tends to infinity. Thus holds a.s., as tends to infinity, and (37) follows.
Next we will prove the original claim. Denote that and . From the previous discussion we know that when , It turns into show that the difference tends to zero a.s. as tends to infinity.
Since is increasing with respect to and , we can get Notice that . We can get, by using the first part of this proof, as tends to infinity because . Similarly the last two summands II and III also tend to zero and we have that when , holds a.s., and the lemma follows.

We can prove the main result in this paper.

Proof of Theorem 2. We prove this theorem by using Theorem 4. We first have to prove that the finite-dimensional distributions of converge to those of .
For any and , we want to show that the the linear combination converges in distribution to a normally distributed random variable with expectation zero and variance The fact that the expectation is zero is trivial. Let us write as
The Lindeberg condition is satisfied if for any , we have as tends to infinity. Consider the set We get an upper bound to by noticing that is increasing with respect to and decreasing with respect to .
Consider the following: where and . Thus we obtain Combining this with the Cauchy-Schwartz inequality, we can get that a.s., which deduces that the Lindeberg condition is satisfied: as tends to infinity, because of and for large . Moreover, we have for all . By Lemma 6 and the fact that it follows that the sum converges to So by Theorem 5, the finite-dimensional distributions of converge to those of .
Finally, we need to prove the tightness of the sequence in . Let , . It is easy to prove that provided for some . It follows from the Cauchy-Schwartz inequality that Notice that for any , We see that if . On the other hand if , then either and or and belong to a same subinterval for some integer . Thus the left-hand side of (67) is zero. Therefore, (67) holds for all and by Theorem 4. This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their sincere gratitude to the editor and the anonymous referees for their valuable comments and error corrections. The project is sponsored by Natural Science Foundation of Anhui Province (no. 1408085QA10), NSFC (11171062), and Innovation Program of Shanghai Municipal Education Commission (12ZZ063).