Limit law of the iterated logarithm for 𝐵-valued trimmed sums

Proceedings Mathematical Sciences

Yuyang Qiu³  Yeling Tong¹  Ke-Ang Fu² 

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

关键词: Banach space;    trimmed sums;    the limit law of the iterated logarithm.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Given a sequence of i.i.d. random variables ${X,X_{n};n≥ 1}$ taking values in a separable Banach space $(B,|cdot |)$ with topological dual 𝐵*, let $X^{(r)}_{n}=X_{m}$ if $| X_{m}|$ is the 𝑟-th maximum of ${| X_{k}|; 1≤ k≤ n}$ and $^{(r)}S_{n}=S_{n}-(X^{(1)}_{n}+cdots+X^{(r)}_{n})$ be the trimmed sums when extreme terms are excluded, where $S_{n}=sum^{n}_{k=1}X_{k}$. In this paper, it is stated that under some suitable conditions,$$limlimits_{n→ ∞}frac{1}{sqrt{2log log n}}maxlimits_{1≤ k≤ n}frac{| {}^{(r)}S_{k}|}{sqrt{k}}=𝜎(X)quadext{a.s.,}$$where $𝜎^{2}(X)=sup_{fin B^{*}_{1}}ext{sf E}f^{2}(X)$ and $B^{*}_{1}$ is the unit ball of 𝐵*.

【 授权许可】

Unknown