Almost sure limit theorems on Wiener chaos: the non-central case

Electronic Communications in Probability

Ehsan Azmoodeh¹ 

关键词: almost sure limit theorem;    multiple Wiener-Itô integrals;    Malliavin calculus;    characteristic function;    Wiener chaos;    Hermite distribution;    fractional Brownian motion;   

DOI  :  10.1214/19-ECP212

学科分类：统计和概率

来源: Institute of Mathematical Statistics

【 摘 要 】

In [1], a framework to prove almost sure central limit theorems for sequences $(G_n)$ belonging to the Wiener space was developed, with a particular emphasis of the case where $G_n$ takes the form of a multiple Wiener-Itô integral with respect to a given isonormal Gaussian process. In the present paper, we complement the study initiated in [1], by considering the more general situation where the sequence $(G_n)$ may not need to converge to a Gaussian distribution. As an application, we prove that partial sums of Hermite polynomials of increments of fractional Brownian motion satisfy an almost sure limit theorem in the long-range dependence case, a problem left open in [1].

【 授权许可】

CC BY