strong law of large numbers

A sequence of random variables $X_1,X_2,\mathrm{\dots }$ with finite expectationsin 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.