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].
Citation
Deli Li. Zhiquan Wu. "The Law of the Iterated Logarithm for $B$-Valued Random Variables with Multidimensional Indices." Ann. Probab. 17 (2) 760 - 774, April, 1989. https://doi.org/10.1214/aop/1176991425
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