Asymptotic behavior for quadratic variations of non-Gaussian multiparameter Hermite random fields

Abstract

Let $(Z_t^{q,H}{)}_{t\in [0,1{]}^d}$ denote a $d$-parameter Hermite random field of order $q\ge 1$ and self-similarity parameter $H=(H_1,\dots ,H_d)\in (\frac{1}{2},1{)}^d$. 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 $(q=1,d\ge 2)$, Rosenblatt process ($q=2$, $d=1$) as well as Rosenblatt sheet $(q=2,d\ge 2)$. For any $q\ge 2,d\ge 1$ and $H\in (\frac{1}{2},1{)}^d$ we show in this paper that a proper normalization of the quadratic variation of $Z^{q,H}$ converges in $L^2(\Omega )$ to a standard $d$-parameter Rosenblatt random variable with self-similarity index $H"=1+(2H-2)/q$.

References in corpus (1)

• Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion