## The Annals of Probability

### The Law of the Iterated Logarithm for $B$-Valued Random Variables with Multidimensional Indices

#### Abstract

Given independent identically distributed random variables $\{X, X_{\bar{n}}; \bar{n} \in \mathbb{N}^d\}$ indexed by $d$-tuples of positive integers and taking values in a separable Banach space $B$ we approximate the rectangular sums $\{\sum_{\bar{k}} \leq \bar{n} X_{\bar{k}}; \bar{n} \in \mathbb{N}^d\}$ by a Brownian sheet and obtain necessary and sufficient conditions for $X$ to satisfy, respectively, the bounded, compact and functional law of the iterated logarithm when $d \geq 2$. These results improve, in particular, the previous work by Morrow [17].

#### Article information

Source
Ann. Probab., Volume 17, Number 2 (1989), 760-774.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176991425

Digital Object Identifier
doi:10.1214/aop/1176991425

Mathematical Reviews number (MathSciNet)
MR985388

Zentralblatt MATH identifier
0679.60007

JSTOR
Li, Deli; Wu, Zhiquan. The Law of the Iterated Logarithm for $B$-Valued Random Variables with Multidimensional Indices. Ann. Probab. 17 (1989), no. 2, 760--774. doi:10.1214/aop/1176991425. https://projecteuclid.org/euclid.aop/1176991425