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

Abstract

Given a sequence of i.i.d. random variables {X,X _n ;n≥1} taking values in a separable Banach space (B, ∥⋅∥) with topological dual B ^∗, let $X_{n}^{(r)}=X_{m}$ if ∥X _m ∥ is the r-th maximum of {∥X _k ∥;1≤k≤n} and ${}^{(r)}S_{n}=S_{n}-(X_{n}^{(1)}+\cdots +X_{n}^{(r)})$ be the trimmed sums when extreme terms are excluded, where $S_{n}={\sum }_{k=1}^{n}X_{k}$. In this paper, it is stated that under some suitable conditions,

$\lim _{n\to \infty }\frac {1}{\sqrt {2\log \log n}}\max _{1\le k\le n}\frac {\|^{(r)}S_{k}\|}{\sqrt {k}}=\sigma (X)~~~\mathrm {a.s.},$

where $\sigma ^{2}(X)=\sup _{f\in B_{1}^{*}}\text {\textsf {E}} f^{2}(X)$ and $B_{1}^{*}$ is the unit ball of B ^∗.