An almost sure invariance principle for some classes of non-stationary mixing sequences

Abstract

In this note we (in particular) prove an almost sure invariance principle (ASIP) for non-stationary and uniformly bounded sequences of random variables which are exponentially fast ϕ-mixing. The obtained rate is of order o(Vn14+δ) for an arbitrary δ>0, where Vn is the variance of the underlying partial sums Sn. For certain classes of inhomogeneous Markov chains we also prove a vector-valued ASIP with similar rates.