Suppose $X_1^{(n)}, X_2^{(n)}...$ are independent and identically distributed random variable with zero mean and $E[|X_1^{(n)}|]<\infty$ for each $n$. Are these conditions sufficient to ensure $$ \sum_{i=1}^n X_i^{(n)} \to 0 $$ almost surely (or in probability)?
If I understand correctly, if we additionally knew that $var(X_1^{(n)})<\infty$ then we could apply Kolmogrov's variance criterion and deduce almost sure convergence.