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.

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