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

概要

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Let {BtH,t ≥ 0} be a fractional Brownian motion (fBm) with Hurst index H ∈ (1/2,1) and let {ξn,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 Un(m,H,f), n ≥ 1 by Un(m,H,f) ≡ n-mH ∑ 0 ≤ j1, j2, …, jm < ∞ f (j1/n, j2/n, …, jm/n)ξj1ξj2…ξjm,where f : R+m → R is a deterministic function. Then the convergence Un(m,H,f) →d ∫R+m f (t1, t2,…, tm)dBt1H dBt2H … dBtmH (n → ∞) is proved to hold for every integer m ≥ 1 under suitable conditions.
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