strong law of large numbers

A sequence of random variables $X_1,X_2,\mathrm{\dots }$ with finite expectations in a probability space is said to satisfiy the strong law of large numbers if

$$\frac{1}{n}\sum _{k=1}^n(X_k-\mathrm{E}[X_k])\stackrel{a.s.}{\to }0,$$

where $a.s.$ stands for convergence almost surely.

When the random variables are identically distributed, with expectation $\mu$, the law becomes:

$$\frac{1}{n}\sum _{k=1}^n{X}_k\stackrel{a.s.}{\to }\mu .$$

Kolmogorov’s strong law of large numbers theorems give conditions on the random variables under which the law is satisfied.

Title	strong law of large numbers
Canonical name	StrongLawOfLargeNumbers
Date of creation	2013-03-22 13:13:10
Last modified on	2013-03-22 13:13:10
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	11
Author	Koro (127)
Entry type	Definition
Classification	msc 60F15
Related topic	MartingaleProofOfKolmogorovsStrongLawForSquareIntegrableVariables