There are a number of questions here about the applicability of the LLN to a sequence of independent Bernoulli random variables $X_n \sim B(p_n)$ when $p_n \to p$. What happens if the sequence $p_n$ does not converge to any $p$?
It seems to me that in this case the sequence is non-ergodic since we only ever see a single draw from $B(p_n)$ and thus we can't even learn what $E[X_n]$ is.
Does it mean then that in this case the LLN does not hold?
