Hermite variations of the fractional Brownian sheet (Q2905264)

A fractional Brownian sheet \((W_{s,t}^{\alpha,\beta})_{(s,t) \in[0,1]^2}\) with Hurst indices \((\alpha,\beta) \in(0,1)^2\) is a centered two-parameter Gaussian process whose covariance function is NEWLINE\[NEWLINE R^{\alpha,\beta}((s_1,t_1), (s_2,t_2)) = \frac12(s_1^{2\alpha} + s_2^{2\alpha} - |s_1-s_2|^{2\alpha}) \frac12(t_1^{2\beta} + t_2^{2\beta} - |t_1-t_2|^{2\beta}) . NEWLINE\]NEWLINE A Hermite variation of order \(q \geq 1\) of a fractional Brownian sheet is NEWLINE\[NEWLINE V_{N,M} = \sum_{i=0}^{N-1} \sum_{j=0}^{M-1} H_q\left(N^{\alpha} M^{\beta} \left( W^{\alpha,\beta}_{\frac{i+1}{N}, \frac{j+1}{M} } - W^{\alpha,\beta}_{\frac{i}{N}, \frac{j+1}{M} } - W^{\alpha,\beta}_{\frac{i+1}{N}, \frac{j}{M} } + W^{\alpha,\beta}_{\frac{i}{N}, \frac{j}{M} }\right) \right) , NEWLINE\]NEWLINE where \(H_q\) is the Hermite polynomial of order \(q\). The main result of the paper is concerned with a limit in law of \(V_{N,M}\) with suitable normalization as \(N,M \to \infty\).NEWLINENEWLINEIt turns out that there are two cases:NEWLINENEWLINE(1) when \(0 < \alpha \leq 1 - \frac{1}{2q}\) or \(0 < \beta \leq 1 - \frac{1}{2q}\), a central limit theorem holds, that is, the limit law is standard normal;NEWLINENEWLINE(2) when \(1 - \frac{1}{2q} < \alpha, \beta < 1\), the limit law is not normal, but the law of a two-parameter Hermite process taken at time \((1,1)\).NEWLINENEWLINEThe main tool used in the paper is Malliavin calculus. The exact normalizations are also given.