Proceedings Mathematical Sciences | |
Precise Asymptotics for Complete Moment Convergence in Hilbert Spaces | |
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 < ð‘ < 2 and $r>1+p/2$,egin{multline*}limlimits_{ðœ€searrow 0}ðœ€^{(2r-p-2)/(2-p)}sumlimits^∞_{n=1}n^{r/p-2-1/p}E{|S_n|-ðœŽðœ€ n^{1/p}}+ =ðœŽ^{-(2r-2-p)/(2-p)}frac{p(2-p)}{(r-p)(2r-p-2)}E|Y|^{2(r-p)/(2-p)},end{multline*}where 𑌠is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator ð›´ , and ðœŽ2 is the largest eigenvalue of ð›´ .
【 授权许可】
Unknown
【 预 览 】
Files | Size | Format | View |
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RO201912040506975ZK.pdf | 196KB | download |