Probability and Mathematical Statistics: Vol. 39, Fasc. 2

ASYMPTOTIC BEHAVIOR FOR QUADRATIC VARIATIONS OF NON-GAUSSIAN MULTIPARAMETER HERMITE RANDOM FIELDS

T. T. Diu Tran

Abstract: Let $(Zq,H ) d t t∈[0,1]$ denote a $d$ -parameter Hermite random field of order $q ≥ 1$ and self-similarity parameter $1 d H = (H1, ...,Hd ) ∈ (2,1)$ . 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 sheet $(q = 2,d ≥ 2)$ . For any $q ≥ 2,d ≥ 1$ and $1 d H ∈ (2,1)$ 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 $H′′ = 1+ (2H - 2)∕q$ .