The Strong Law of Large Numbers without Recycling

Calvin Wooyoung ChinJanuary 21, 2021January 23, 2021NotesLaw of large numbers, Probability, Symmetrization

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 $EX_{11}=0$ .