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.