# Proving convergence almost surely

I need to prove that $S_n := \frac{1}{n}(X_1X_2 + X_2X_3 +...+X_nX_{n+1})$, where $X_k$ are rvs independtly Poisson distributed with parameter $\lambda = 2$ for any natural $k$, converges almost surely.

I proved that this sequence converges in probability to 4 because of The Weak Law of Large Numbers. But I'm struggling with a.s. convergence. I thought about Kolmogorov 0-1 theorem but it leads me nowhere.

• Maybe try splitting $S_n$ into two sums of even and odd terms, then apply SLLN to each? – Mike Earnest Aug 23 '18 at 17:29
• @MikeEarnest Yes, that was the key. Thanks for help – treskov Aug 23 '18 at 17:45