Asymptotic Behavior for Quadratic Variations of Non-Gaussian Multiparameter Hermite Random Fields

Let (Ztq,H)t∈[0,1]d denote a d-parameter Hermite random field of order q ≥ 1 and self-similarity parameter H = (H1,…, Hd) ∈ (1/2, 1)d. This process is H-self-similar, has stationary increments and exhibits long-range dependence. Particular examples include fractional Brownian motion (q = 1, d = 1), fractional Brownian sheet (q = 1, d ≥ 2), the Rosenblatt process (q = 2, d = 1) as well as the Rosenblatt sweet (q = 2, d ≥ 2). For any q ≥ 2, d ≥ 1 and H ∈ (1/2, 1)d we show in this paper that a proper renormalization of the quadratic variation of Zq,H converges in L2(Ω) to a standard d-parameter Rosenblatt random variable with self-similarity index Hʺ = 1 + (2H − 2)/q.

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Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).