Behavior of the generalized Rosenblatt process at extreme critical exponent values

Abstract

The generalized Rosenblatt process is obtained by replacing the single critical exponent characterizing the Rosenblatt process by two different exponents living in the interior of a triangular region. What happens to that generalized Rosenblatt process as these critical exponents approach the boundaries of the triangle? We show by two different methods that on each of the two symmetric boundaries, the limit is non-Gaussian. On the third boundary, the limit is Brownian motion. The rates of convergence to these boundaries are also given. The situation is particularly delicate as one approaches the corners of the triangle, because the limit process will depend on how these corners are approached. All limits are in the sense of weak convergence in C[0, 1]. These limits cannot be strengthened to convergence in L 2 (Ω).

Generalized k-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus

“…In this part, we restrict ourselves to the case of quadratic variations (i. e. k = 2) and we derive a multidimensional CLT. This result will be needed in Section 5 which deals with the estimation of the Hurst parameter of the solution to (2). To establish the multidimensional convergence, we will use Theorem 6.2.3 in [21].…”

“…In Section 4, we show a non-central limit limit in the case k = 2, H > 3 4 and for filters of order p = 1. Section 5 concerns the estimation of the Hurst parameter of the solution to the fractional-white wave equation (2). We included here theoretical results related to the behavior of the kvariations estimators for the Hurst index as well as simulations and numerical analysis for the performance of the estimators.…”

Limit behavior of the Rosenblatt Ornstein–Uhlenbeck process with respect to the Hurst index

“…There are several recent research works that investigate the asymptotic behavior in distribution of some fractional processes (see [3], [2], [1], [15]) with respect to the Hurst parameter. In particular, in the case of the Rosenblatt process (Z H (t)) t≥0 with self-similarity index H ∈ ( 1 2 , 1), it has been shown in [15] that Z H converges weakly, as H → 1 2 , in the space of continuous functions C[0, T ] (for every T > 0), to a Brownian motion while if H → 1, it tends weakly to the stochastic process (t 1 √ 2 (Z 2 − 1)) t≥0 , Z 2 − 1 being a so-called centered chi-square random variable.…”

“…In particular, in the case of the Rosenblatt process (Z H (t)) t≥0 with self-similarity index H ∈ ( 1 2 , 1), it has been shown in [15] that Z H converges weakly, as H → 1 2 , in the space of continuous functions C[0, T ] (for every T > 0), to a Brownian motion while if H → 1, it tends weakly to the stochastic process (t 1 √ 2 (Z 2 − 1)) t≥0 , Z 2 − 1 being a so-called centered chi-square random variable. The case of the generalized Rosenblatt process has been considered in [2] while the case of the Rosenblatt sheet can be found in [1]. Hermite processes of higher order have been considered in [3], [1].…”

Behavior with respect to the Hurst index of the Wiener Hermite integrals and application to SPDEs

“…Our work continues a recent line of research that concerns the limit behavior in distribution with respect to the Hurst parameter of Hermite and related fractionaltype stochastic processes. In particular, the papers [5] and [2] deal with the asymptotic behavior of the generalized Rosenblatt process, the work [1] studies the multiparamter Hermite processes while the paper [22] investigates the Ornstein-Uhlenbeck process with Hermite noise of order q = 2.…”