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].