A series expansion of fractional Brownian motion
成果类型：
Article
署名作者：
Dzhaparidze, K; van Zanten, H
署名单位：
Vrije Universiteit Amsterdam
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-003-0310-2
发表日期：
2004
页码：
39-55
关键词：
摘要：
Let B be a fractional Brownian motion with Hurst index His an element of(0,1). Denote by x(1)<... the positive, real zeros of the Bessel function J(-H) of the first kind of order -H, and let y(1),y(2)<... be the positive zeros of J(1-H). In this paper we prove the series representation [GRAPHICS] where X-1,X-2,... and Y-1,Y-2,... are independent, Gaussian random variables with mean zero and VarX(n)=2c(H)(2)x(n)(-2H)J(1-H)(-2)(x(n)), VarY(n)=2c(H)(2)y(n)(-2H)J(-H)(-2)(y(n)), and the constant c(H)(2) is defined by c(H)(2)=pi(-1)Gamma(1+2H) sin piH. We show that with probability 1, both random series converge absolutely and uniformly in tis an element of[0,1], and we investigate the rate of convergence.