On the Cluster Set Problem for the Generalized Law of the Iterated Logarithm in Euclidean Space

Abstract

In a recent paper by the author it has been shown that there exists a general law of the iterated logarithm (LIL) in Banach space, which contains the LIL of Ledoux and Talagrand and an LIL for infinite-dimensional random variables in the domain of attraction to a Gaussian law as special cases. We now investigate the corresponding cluster set problem, which we completely solve for random vectors in two-dimensional Euclidean space. Among other things, we show that all cluster sets arising from this generalized LIL must be sets of diameter 2, which are star-shaped and symmetric about the origin, and any closed set of this type occurs as a cluster set for a suitable random vector. Moreover, we show that if the random vectors under consideration have independent components, one only obtains cluster sets from the subclass of all sets, which can be represented as closures of countable unions of standard ellipses.