A series expansion of fractional Brownian motion

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 J_1–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 c_H² is defined by c_H²=^–1(1+2H) sin H. We show that with probability 1, both random series converge absolutely and uniformly in t0,1], and we investigate the rate of convergence.Mathematics Subject Classification (2000): 60G15, 60G18, 33C10