Escape rates for multi-dimensional shift selfsimilar additive sequences

First the relation between shift selfsimilar additive sequences and stationary sequences of Ornstein-Uhlenbeck type (OU type) on ${\mathbb{R}}^d$ is shown and then the rates of escape for shift selfsimilar additive sequences are discussed. As a corollary, fundamental problems on recurrence of stationary sequences of OU type are solved. Some applications to laws of the iterated logarithm for strictly stable L'evy processes on ${\mathbb{R}}^d$ and independent Brownian motions are given.