## The Annals of Probability

- Ann. Probab.
- Volume 22, Number 2 (1994), 1026-1043.

### On Strassen's Law of the Iterated Logarithm in Banach Space

#### Abstract

Let $\{X, X_n; n \geq 1\}$ be a sequence of i.i.d. random variables with values in a separable Banach spacc $B$ and set, for each $n, S_n = X_1 + \cdots + X_n$. We give necessary and sufficient conditions in order that $\lim\sup_{n\rightarrow\infty} n^{-1-(p/2)}(2L_2n)^{-(p/2)}\sum_{i=1}^n\|S_i\|^p < \infty \mathrm{a.s.},$ $\lim\sup_{n\rightarrow\infty} n^{-1-(p/2)}(2L_2n)^{-(p/2)}\sum_{i=0}^n\|S_n - S_i\|^p < \infty \mathrm{a.s.},$ where $p \geq 1$. Furthermore, the exact values of the above $\lim \sup$ are obtained. Some results are the extensions of Strassen's work to the vector settings and some are new even on the real line. The proofs depend on the construction of an independent sequence with values in $l_p(B)$ and appear as an illustration of the power of the limit law in Banach space.

#### Article information

**Source**

Ann. Probab., Volume 22, Number 2 (1994), 1026-1043.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988739

**Digital Object Identifier**

doi:10.1214/aop/1176988739

**Mathematical Reviews number (MathSciNet)**

MR1288141

**Zentralblatt MATH identifier**

0805.60006

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

Secondary: 60F15: Strong theorems

**Keywords**

Law of the iterated logarithm Banach space type 2 Banach space

#### Citation

Chen, Xia. On Strassen's Law of the Iterated Logarithm in Banach Space. Ann. Probab. 22 (1994), no. 2, 1026--1043. doi:10.1214/aop/1176988739. https://projecteuclid.org/euclid.aop/1176988739