I'm wondering about a SLLN where the number of summands is also allowed to be a random variable. More specifically assume that:
- $\{X_i\}$ are iid with $\mathbb{E}[|X_1|] < \infty $
- $N=\{N_n\}_{n\in\mathbb{N}}$ is a sequence of random positive integers independent of $\{X_i\}$
- $\lim_{n} N_n =\infty$ $a.s$
Can we conclude that $ \frac{1}{N_n} \sum_{i=1}^{N_n} X_i = \mathbb{E}[X_1] $ almost surely?