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.

The random variables are identically distributed;
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
Canonical name	KolmogorovsStrongLawOfLargeNumbers
Date of creation	2013-03-22 13:13:12
Last modified on	2013-03-22 13:13:12
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	7
Author	Koro (127)
Entry type	Theorem
Classification	msc 60F15
Synonym	Kolmogorov’s criterion
Related topic	MartingaleProofOfKolmogorovsStrongLawForSquareIntegrableVariables
Related topic	ProofOfKolmogorovsStrongLawForIIDRandomVariables