2016/07/27 by Tran, T. T. Diu
#60F05 #60G22 #60H05 #60H07 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1607.08278
Let (Zt(q, H))t ≥ 0 denote a Hermite process of order q ≥ 1 and self-similarity parameter H ∈ ((1)/(2), 1). Consider the Hermite-driven moving average process Xt(q, H) = ∫0t x(t-u) dZ(q, H)(u), t ≥ 0. In the special case of x(u) = e-θu, θ> 0, X is the non-stationary Hermite Ornstein-Uhlenbeck process of order q. Under suitable integrability conditions on the kernel x, we prove that as T → ∞, the normalized quadratic functional GT(q, H)(t)=\frac1T2H0 - 1∫0Tt((Xs(q, H))2 - E[(Xs(q, H))2]) ds , t ≥ 0, where H0 = 1 + (H-1)/q, converges in the sense of finite-dimensional distribution to the Rosenblatt process of parameter H' = 1 + (2H-2)/q, up to a multiplicative constant, irrespective of self-similarity parameter whenever q ≥ 2. In the Gaussian case (q=1), our result complements the study started by Nourdin et al in arXiv:1502.03369, where either central or non-central limit theorems may arise depending on the value of self-similarity parameter. A crucial key in our analysis is an extension of the connection between the classical multiple Wiener-Itô integral and the one with respect to a random spectral measure (initiated by Taqqu (1979)), which may be independent of interest.