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

摘要

Let (Zq,Ht )t2[0,1]d denote a d-parameter Hermite randomfield of order q - 1 and self-similarity parameter H = (H1, . . . ,Hd) 2( 12 , 1)d. This process is H-self-similar, has stationary increments and exhibitslong-range dependence. Particular examples include fractional Brownianmotion (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 H 2( 12 , 1)d we show in thispaper that a proper renormalization of the quadratic variation of Zq,H convergesin L2(Ω) to a standard d-parameter Rosenblatt random variable withself-similarity index H00= 1 + (2H ? 2)/q.