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 ∗.