I know the WLLN and SLLN applies for iid random variables, but it seems like it should also hold for a sequence of simply independent Bernouli 0/1 flips.
So we have $\{X_n\}$ where each has $EX_n=\Pr(X_n=1)=p_n$.
I've found some generalizations of the LLNs, but am unsure which might apply. Does one exist and what's a good reference? Will I need any kind of conditions like a finite number of distinct $p_n$s?
You can apply the variance criterion for averages (c.f. corollary 4.22 in Kallenberg). It states the following:
If $\xi_1, \xi_2, \ldots$ is a sequence of independent random variables with mean $0$ and $\sum \limits_{n = 1}^\infty n^{-2c} E[\xi_n^2] < \infty$ for some $c > 0$, then $n^{-c} \sum \limits_{i = 1}^n \xi_i$ converges to $0$ almost surely.
Most notably, if your random variables $X_i$ have a uniformly bounded variance (or if they are more specifically uniformly bounded), you may choose $c = 1$ and $\xi_i = X_i - E[X_i]$. Then the theorem shows that $\frac{1}{n} \sum \limits_{i = 1}^n\left( X_i - E[X_i]\right)$ converges almost surely to $0$.