Recurrence Sets of Normed Random Walk in $R^d$

Abstract

In this paper we give examples of the sets of recurrent points (or accumulation points) of random walks in $R^d$ normed by nice sequences of constants. These examples, interesting in their own right, give rise to some very interesting conjectures concerning the general structure of such sets. Of particular interest are the recurrent points of the ordinary averages or sample means. It turns out that any closed subset of $R^d$ can be the finite points of recurrence of a sequence of averages: $(X_1 + \cdots + X_n)/n, X_i$ i.i.d. random vectors. This seems to be a property not shared by most other normalizing sequences. We also give some results on rates of escape of random walks in a domain of attraction. In looking for rates of escape we are looking for normalizing constants which give rise to no finite recurrent points of the normalized walk.