# The Strong Law of Large Numbers without Recycling

The strong law of large numbers heavily depends on the fact that each of $X_1,X_2,\ldots$ appears repeatedly in the sequence $X_1,(X_1+X_2)/2, (X_1+X_2+X_3)/3,\ldots$. We prove a version of the strong law of large number in which no random variable gets “recycled.” More precisely, we prove the Hsu-Robbins-Erdős theorem, which states that for i.i.d. $\{X_{ni}\}$, we have $(X_{n1}+\cdots+X_{nn})/n \to 0$ if and only if $EX_{11}^2 < \infty$ and $EX_{11}=0$.