# Kolmogorov’s strong law of large numbers

Let $X_{1},X_{2},\dots$ be a sequence of independent random variables, with finite expectations. The strong law of large numbers holds if one of the following conditions is satisfied:

1. 1.

The random variables are identically distributed;

2. 2.

For each $n$, the variance of $X_{n}$ is finite, and

 $\sum_{n=1}^{\infty}\frac{\operatorname{Var}[X_{n}]}{n^{2}}<\infty.$
Title Kolmogorov’s strong law of large numbers KolmogorovsStrongLawOfLargeNumbers 2013-03-22 13:13:12 2013-03-22 13:13:12 Koro (127) Koro (127) 7 Koro (127) Theorem msc 60F15 Kolmogorov’s criterion MartingaleProofOfKolmogorovsStrongLawForSquareIntegrableVariables ProofOfKolmogorovsStrongLawForIIDRandomVariables