Limiting behaviour of pattern counts in biased binary strings

For $p \in (0,1)$, sample a binary sequence from the infinite product measure of Bernoulli$(p)$ distributions.

It is known that for $p=1/2$, almost every binary sequence is Poisson generic in the sense of Peres and Weiss, a property that reflects a specific statistical pattern in the frequency of finite substrings.

However, this behaviour is highly exceptional: it fails for any $p \ne 1/2$.

In these other cases, we show that the frequency of substrings of almost every sequence has either trivial or peculiar behaviour.

Nevertheless, the Poisson limiting regime can be recovered if one restricts attention to substrings with a fixed number of successes in the Bernoulli$(p)$ trials.