A threshold for Poisson behavior of non-stationary product measures

Let ${\gamma }_n=O({\log }^{-c}n)$ and let $\nu$ be the infinite product measure whose $n$-th marginal is Bernoulli$(1\mathrm{/}2+{\gamma }_n)$. We show that $c=1\mathrm{/}2$ is the threshold, above which $\nu$-almost every point is simply Poisson generic in the sense of Peres-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.