  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.
