A threshold for Poisson behavior of non-stationary product measures

Let γ ₙ= O (log ⁻ᶜn) and let ν be the infinite product measure whose n th marginal is Bernoulli (1/2+γ ₙ) . We show that $c=1/2$ is the threshold, above which ν -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 ν is singular with respect to the uniform product measure, but ν -almost every point is simply Poisson generic.