Suppose that we have an array of identically distributed real valued random variables $((X_{i,n})_{i=1}^n)_{n\in\mathbb{N}}$ such that for each $n\in\mathbb{N}$, $X_{1,n},X_{2,n},..,X_{n,n}$ are independent. Assume that $\mathbb{E}[X_{i,n}]=0$.
I would like to say that
$\lim_{n\rightarrow\infty}n^{-1}\sum_{i=1}^nX_{i,n}=0$ almost surely.
Presumably the first moment assumption is not sufficient to guarantee almost sure convergence? But if one assumes that the fourth moment of $X_{i,n}$ is finite then the result does hold. classic proofs of the strong law of large numbers assuming a finite fourth moment carry over to this triangular array result with no change.
The question is: what if we only assume a finite second moment? Does the almost sure convergence hold?