Precise Asymptotics for Complete Moment Convergence in Hilbert Spaces

Proceedings Mathematical Sciences

Juan Chen¹  Keang Fu² 

[1] $$;School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 00, China$$

关键词: Complete convergence;    complete moment convergence;    convergence rates;    Hilbert spaces;    precise asymptotics.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Let ${X, X_n;n≥ 1}$ be a sequence of i.i.d. random variables taking values in a real separable Hilbert space $(H,|cdot p|)$ with covariance operator $sum$. Set $S_n=sum^n_{i=1}X_i,n≥ 1$. We prove that for 1 < 𝑝 < 2 and $r>1+p/2$,egin{multline*}limlimits_{𝜀searrow 0}𝜀^{(2r-p-2)/(2-p)}sumlimits^∞_{n=1}n^{r/p-2-1/p}E{|S_n|-𝜎𝜀 n^{1/p}}+ =𝜎^{-(2r-2-p)/(2-p)}frac{p(2-p)}{(r-p)(2r-p-2)}E|Y|^{2(r-p)/(2-p)},end{multline*}where 𝑌 is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator 𝛴 , and 𝜎2 is the largest eigenvalue of 𝛴 .

【 授权许可】

Unknown