INTEGRATED FUNCTIONALS OF NORMAL AND FRACTIONAL PROCESSES

• 成果类型: Article
• 署名作者: Buchmann, Boris; Chan, Ngai Hang
• 署名单位: Monash University; Chinese University of Hong Kong
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/08-AAP531
• 发表日期: 2009
• 页码: 49-70
• 关键词: CENTRAL LIMIT-THEOREMS; stochastic difference-equations; stationary gaussian-processes; brownian-motion; nonlinear functionals; CONVERGENCE; FIELDS

摘要

Consider Z(t)(f) (u) = integral o(tu) f (N(s)) ds, t > 0, u is an element of [0, 1], where N = (N(t))t is an element of R is a normal process and f is a measurable real-valued function satisfying Ef(N(0))(2) < infinity and Ef (N(0)) = 0. If the dependence is sufficiently weak Hariz [J. Multivariate Anal. 80 (2002) 191-216] showed that Z(t)(f)/t(1/2) converges in distribution to a multiple of standard Brownian motion as t -> infinity. If the dependence is sufficiently strong, then Zt/(EZ(t) (1)(2))(1/2) converges in distribution to a higher order Hermite process as t -> infinity by a result by Taqqu [Wahrsch. Verw. Gebiete 50 (1979) 53-83]. When passing from weak to strong dependence, a unique situation encompassed by neither results is encountered. In this paper, we investigate this situation in detail and show that the limiting process is still a Brownian motion, but a nonstandard norming is required. We apply our result to some functionals of fractional Brownian motion which arise in time series. For all Hurst indices H E (0, 1), we give their limiting distributions. In this context, we show that the known results are only applicable to H < 3/4 and H > 3/4, respectively, whereas our result covers H = 3/4.