A threshold for Poisson behavior of non-stationary product measures

Abstract Let $\gamma _{n}= O (\log ^{-c}n)$ and let $\nu $ be the infinite product measure whose n th marginal is Bernoulli $(1/2+\gamma _{n})$ . We show that $c=1/2$ is the threshold, above which $\nu $ -almost every point is simply Poisson generic in the sense of Peres and Weiss, and below which this can fail. This provides a range in which $\nu $ is singular with respect to the uniform product measure, but $\nu $ -almost every point is simply Poisson generic.