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

Escape rates for multi-dimensional shift selfsimilar additive sequences