Tempered Hermite process

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

In this paper, we introduce a new class of stochastic processes, which we call tempered Hermite processes. Tempered Hermite processes modify the kernel of Z k H by multiplying an exponential tempering factor λ > 0 such that they are well defined for Hurst parameter H > 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 are defined for all functions f ∈ L 2 (R k ). The prime ′ on the integral sign shows that one does not integrate on the diagonals x i = x j , i = j. See, for example, [12,Chapter 4].
where (x) + = xI(x > 0) and d = 1 , is called a tempered Hermite process of order k.
The next lemma shows that Z k H,λ (t), given by (2), is well defined for any t ≥ 0. Lemma 1. The function is well defined in L 2 (R k ) for any H > 1 2 and λ > 0.
where we applied the standard integral formula [13, p. 344 for | arg β| < π, Re µ > 0, Re ν > 0. Here K ν (x) is the modified Bessel function of the second kind (see [1,Chapter 9]). Next, we need to show that which is finite, and, consequently, from (4) and (6) we get where we applied the fact that K ν (z) = K −ν (z) and k ν (z) < z −ν 2 ν−1 Γ (ν) for z > 0. Hence, from (4) and (7) it follows that Since K 0 (z) ∼ − log(z) as z → 0 (see [1, Eq. 9.6.8], we have Now, we find an upper bound for I 2 . It can be shown that Kν (x) Kν (y) > e y−x for 0 < x < y and an arbitrary real number ν (see [4]). Therefore, From (8), (9), and (10) we can see that for ǫ > 0, and this shows that The next result shows that although a tempered Hermite process is not a selfsimilar process, it does have a nice scaling property. Here the symbol indicates the equivalence of finite-dimensional distributions. Proposition 1. The process Z k H,λ given by (2) has stationary increments such that for any scale factor c > 0.
Proof. Since B(dy) has the control measure m(dy) = σ 2 dy, the random measure B(c dy) has the control measure c 1/2 σ 2 dy. Given t j , j = 1, . . . , n, by the change of variables s = cs ′ and y i = cy so that (12) holds. Suppose now that s j < t j and change the variables 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 Z k H,λ (t) has a continuous version. Proof. According to the proof of Lemma 1, where c 1 and c 2 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 Apply (14) for tempered Hermite process Z k H,λ (t) by taking p = 1, β = min{1, 2H − 1}, and c = min{c 1 , c 2 } to get the desired result.
We next compute the covariance function of Z k H,λ (t). Unlike Hermite processes, the covariance function of a tempered Hermite process is different for different k ≥ 1.

Proposition 3.
The process Z k H,λ given by (2) has the covariance function for λ > 0 and d > 1 2 − 1 2k (equivalently, H > 1 2 ). Proof. By applying the Fubini theorem and the isometry of multiple Wiener-Itô integrals we have for any H > 1 2 and λ > 0, and hence we get the desired result. Let where B(·) is a complex-valued Gaussian random measure, and C H,k = ( ) k is a constant depending on H and k. The double prime ′′ on the integral indicates that one does not integrate on the diagonals ω i = ω j , i = j.

Proof. We first observe that
has the Fourier transform 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 which is equivalent to (15).

Limit theorem
In this section, we show that the process Z k H,λ (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, the discrete process Y λ,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 Y λ,k (n), where the limit involves a fractional Brownian motion. When k > 1 and λ = 0, Taqqu [26] showed that the weak convergence limit of Y λ,k (n) is the Hermite process (1).
The following proposition gives a powerful tool for proving the result of this section. 17) for N = 1, 2, . . ., where g N ∈ L 2 (Z k ) for k ≥ 1, and {ε n } is an i.i.d. sequence with mean zero and variance 1. Assume that, for some f ∈ L 2 (R k ),
Define the discrete chaos process where the prime ′ indicates exclusion of the diagonals i p = i q , p = q, {ε n } is as before, and Y λ,k (n), 0 ≤ t ≤ 1.

Theorem 1.
Let Y λ,k (n) be the discrete chaos process given by (18). Then where ⇒ means weak convergence in the Skorokhod space D[0, 1] with uniform metric, Z k H,λ (t) is the tempered Hermite process in (2), and H = 1 + kd − k 2 . Remark 1. The Lamperti's theorem [15] states that if −→ means the convergence of finite-dimensional distributions). In our case, since the stationary processes {Y λ N k } depend on N through the parameter λ, the limit process {Z k H,λ (t)} t≥0 need not be a self-similar process. Therefore, the result of Theorem 1 does not contradict the Lamperti theorem.
In order to show the tightness, we need to verify that where γ > 0 and α > 1 2 (here {F n } n≥1 is a sequence of nondecreasing continuous functions on [0, 1] that are uniformly bounded and satisfy Therefore, we get Now, we consider two different cases corresponding with the range of d. First, assume for H > 1. Now, it remains to apply (23) by selecting γ = 1, α = min{H, 1}, and F n (t) = t to get the desired result.