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$(1/2+\gamma_{n})$. We show that $c=1/2$ is the threshold, above which $\nu$-almost every point is simply Poisson generic in the sense of Peres-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.