A strong invariance principle for associated sequences

Abstract

By combining the Berkes-Philipp blocking technique and the Csorgo-Revesz quantile transform methods, we find that partial sums of an associated sequence can be approximated almost surely by partial sums of another sequence with Gaussian marginals. A crucial fact is that this latter sequence is still associated with covariances roughly bounded by the covariances of the original sequence, and that one can approximate it by an iid Gaussian process using the Berkes-Philipp method. We require that the original sequence has finite $(2 + r)$th moments, $r > 0$, and a power decay rate of a coefficient $u(n)$ which describes the covariance structure of the sequence. Based on this result, we obtain a strong invariance principle for associated sequences if $u(n)$ exponentially decreases to 0.