# Strong Law of Large Numbers with randomly many summands

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?

Let $$\Omega$$ be the underlying probability space. Define $$Y_n = \frac{1}{n}\sum_{i = 1}^n X_i\,.$$ By the strong law of large numbers, if we let $$\Omega_1 = \{ Y_n \to E[X_1]\}$$ then $$P(\Omega_1) =1$$. Similarly, define $$\Omega_2 = \{N_n \to \infty\}$$; then $$P(\Omega_2) = 1$$. Then $$P(\Omega_1\cap \Omega_2) = 1$$ and on the set $$\Omega_1 \cap \Omega_2$$ we have $$Y_{N_n} \to E[X_1]$$. Note that in fact you don't even need $$N_n$$ to be independent of $$\{X_i\}$$ just that $$N_n \to \infty$$.