Escape Rates for Multidimensional Shift Self-similar Additive Sequences

First the relation between shift self-similar additive sequences and stationary sequences of Ornstein–Uhlenbeck type (OU type) on R d is shown, and then the rates of escape for shift self-similar 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 Lévy processes on R d and independent Brownian motions are given.