The Law of the Iterated Logarithm for Independent Random Variables with Multidimensional Indices

Abstract

Let $X_{\bar n}, \bar{n} \in \mathbb{N}^d$, be a field of independent real random variables, where $\mathbb{N}^d$ is the $d$-dimensional lattice. In this paper, the law of the iterated logarithm is established for such a field of random variables. Theorem 1 brings into focus a connection between a certain strong law of large numbers and the law of the iterated logarithm. A general technique is developed by which one can derive the strong law of large numbers and the law of the iterated logarithm, exploiting the convergence rates in the weak law of large numbers in Theorem 2. In Theorem 3, we use Gaussian randomization techniques to obtain the law of the iterated logarithm which generalizes Wittmann's result.