Approximating the Rosenblatt Process by Multiple Wiener Integrals *

Let Z H be the Rosenblatt process with the representation Z H t = t 0 t 0 L H (t, s, r)dBsdBr, where B is a standard Brownian motion,

L H (t, s, r)dBsdBr, where B is a standard Brownian motion, 1  2 < H < 1 and L H is a given kernel.By reviewing the kernel L H we construct its approximation of multiple Wiener integrals of the form

Introduction
Hermite process is a special class of self-similar processes with long-range dependence.The processes arise from the Non Central Limit Theorem studied by Taqqu [12,13] and Dobrushin-Majòr [7].The famous fractional Brownian motion and Rosenblatt process are its special examples.Let us briefly recall the general context.Let (ξ n ) n∈N be a stationary centered Gaussian sequence with E(ξ where l ≥ 1 is an integer, H ∈ ( 1 2 , 1) and L is a slowly varying function at infinity, and let the Borel function g : R → R satisfy E(g(ξ 0 )) = 0, E(g(ξ 0 ) 2 ) < ∞ and where H j is the Hermite polynomial of order j defined by H j (x) = (−1) j e x 2 2 d j dx j e − x 2 2 , j = 1, 2, . . .with H 0 (x) = 1.Then, the constant l = min{j ; c j = 0}.
Clearly, when l = 1 Hermite process is the fractional Brownian motion with Hurst parameter H ∈ ( 1 2 , 1).When l = 2 the Hermite process is called the Rosenblatt process (see Taqqu [12]).It is important to note that Hermite process is not Gaussian for l ≥ 2. The simplest Hermite process is fractional Brownian motion, and the Rosenblatt process is the simplest non-Gaussian Hermite process.Hermite processes are neither a semi-martingale nor a Markov process, and the following properties hold: (i) they are the long-range dependence in the sense of (ii) they are H-selfsimilar; (iii) they have stationary increments; (iv) they admit the same covariance functions, i.e.
(v) they are Hölder continuous of order γ < H.
These good properties of the Hermite process motivate us to study it.More works for the Hermite process and Rosenblatt process can be found in Bardet et al [3], Chen et al [4], Chronopoulou et al [5,6], Garzón et al [8], Maejima-Tudor [9], Peccati and Taqqu [10], Pipiras-Taqqu [11], Torres-Tudor [14], Tudor [15], Tudor-Viens [16] and the references therein.In this paper we will prove an approximation theorem of Rosenblatt process based on the multiple integrals of form with k 1 , k 2 > 0. For simplicity we denote Z t (H, 2) = Z H t .The motivation to consider the approximation arises from the following estimate: for all t ∈ [0, T ] and s, r > 0. In order to prove the above estimate, without loss of generality, we may assume that s ≥ r and we have by making the substitutions u − s = x(s − r).It follows that for all t ∈ [0, T ] and y 1 , y 2 > 0.
It is important to note that if the above minimum is attained at the function ζ * , then ζ * > 0 a.e.In fact, we have This gives the contradiction.Thus, we may assume that k 1 , k 2 > 0 in (1.4) and study the best approximation problem where When l = 1, Hermite process is a fractional Brownian Motion with Hurst index H and the similar approximation is first considered by Banna-Mishura [1,2].When l ≥ 2, the question has not been studied and this process is non-Gaussian with non-trivial analysis.
In order to state our object, let us consider the kernel K H of the form and β(•, •) denotes the classical Beta function.Then we have (see, for example, Tudor [15]) where d(H) = 1 H+1 (4H − 2)H −1 and H = 1 2 (1 + H).In this short note, our main aim is to find the optimal approximation of Z H t by (1.4) via calculating accurately the values of k 1 , k 2 .In order to end this one can easily check that (see Section 3) is a quadratic polynomial in x = k 1 t −2α and its discriminant is also a quadratic polynomial in k 2 with the discriminant By using the constant D 1 we give our main result and at the end of this paper we give the numerical simulations of these constants (see Figure 1, 2, 3 and Table 1).This note is organized as follows.In Section 2, we give the representation of the In Section 3 and Section 4, we consider the optimal approximation in the two cases D 1 ≤ 0 and D 1 > 0, respectively.In Section 5 we consider two special cases.

The representation of
start with the finiteness of the constant C 2 (H).
Lemma 2.1.For all Proof.By Young's inequality, we have As an immediate result we see that a(k for all 1 2 < H < 1.
Proof of Theorem 2.2.An elementary calculation can show that for all t ∈ [0, T ], which give for all t ∈ [0, T ].On the other hand, it is easy to calculate that for all ζ ∈ K.This completes the proof.
3 The optimal approximation, case D 1 ≤ 0 In order to obtain the optimal approximation in the case D 1 ≤ 0 we need some preliminaries and keep the notation in Section 2. Denote α = H − 1 2 and define the with x = k 1 t −2α .Clearly, the discriminant D of the quadratic polynomial G(x) satisfies This gives a quadratic polynomial in k 2 and its discriminant is D 1 .
is the stagnation point of the function An elementary calculation can obtain .
Proof.This is a simple exercise.In fact, for all 1 2 < H < 1 we have by Cauchy inequality, and it is easy to check that the inequality above is strict.
Proof of Theorem 3.1.Let now D 1 ≤ 0. Then we see that D ≤ 0 and ∂f ∂t ≥ 0 for H ∈ ( 1 2 , 1) be the stagnation point of the function Then (k * 1 , k * 2 ) can be given by (3.3) and (3.4), and elementary calculations may obtain the Hessian matrix H on f (T, k 1 , k 2 ) as follows ) .Combining this with (3.5), we get |H| > 0 for all H ∈ ( 1 2 , 1), which means that the minimal value of (k . Thus, we have proved the theorem.
4 The optimal approximation, case D 1 > 0 In this section we throughout let D 1 > 0 and keep the notation in Section 3 and Section 2. When D 1 > 0, the equation D = 0 admits two real roots as follows . By (4.1) we have that and the theorem follows.
Next we consider the case k ), then D ≤ 0 and we have By solving the equation ∂f (T,k1, * ) k1 = 0, we get the stagnation points of the functions ).
Hence we have ).
On the other hand, we can also get and the inequality (4.2) follows.This completes the proof.
Proof.From (4.5) it follows that On the other hand, (3.1) implies that is a quadratic function in x = t −2α , and by (2.1), we get < T by a simple analysis and Lemma 4.3.
On the other hand, from h(0, k This completes the proof.

Two special cases
In this section we consider two special classes of the approximation functions ζ ∈ K.