A series expansion of fractional Brownian motion

Abstract.Let B be a fractional Brownian motion with Hurst index H∈(0,1). Denote by the positive, real zeros of the Bessel function J−H of the first kind of order −H, and let be the positive zeros of J1−H. In this paper we prove the series representation where X1,X2,... and Y1,Y2,... are independent, Gaussian random variables with mean zero and and the constant cH2 is defined by cH2=π−1Γ(1+2H) sin πH. We show that with probability 1, both random series converge absolutely and uniformly in t∈[0,1], and we investigate the rate of convergence.

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