Convergence of Weighted Sums of Products of Random Variables with Long-Range Dependence

Abstract

Let {B_t^H,t ≥ 0} be a fractional Brownian motion (fBm) with Hurst index H ∈ (1/2,1) and let {ξ_n,n ≥ 0} be a sequence of centered random variables with stationary, long-range dependence increments. For every integer m ≥ 1 we define the random series U_n(m,H,f), n ≥ 1 by
U_n(m,H,f) ≡ n^-mH ∑ _{0 ≤ j1, j2, …, jm < ∞}  f (j₁/n, j₂/n, …, j_m/n)ξ_j1ξ_j2…ξ_jm,
where f : R₊^m → R is a deterministic function. Then the convergence
U_n(m,H,f) →^d  ∫_R+^m  f (t₁, t₂,…, t_m)dB_t1^H dB_t2^H … dB_tm^H     (n → ∞)
is proved to hold for every integer m ≥ 1 under suitable conditions.