# strong law of large numbers

A sequence of random variables $X_{1},X_{2},\dots$ with finite expectations in a probability space is said to satisfiy the if

 $\frac{1}{n}\sum_{k=1}^{n}(X_{k}-\operatorname{E}[X_{k}])\xrightarrow[]{a.s.}0,$

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

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

 $\frac{1}{n}\sum_{k=1}^{n}X_{k}\xrightarrow[]{a.s.}\mu.$

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 StrongLawOfLargeNumbers 2013-03-22 13:13:10 2013-03-22 13:13:10 Koro (127) Koro (127) 11 Koro (127) Definition msc 60F15 MartingaleProofOfKolmogorovsStrongLawForSquareIntegrableVariables