Famous Theorems of Mathematics/Law of large numbers

< Famous Theorems of Mathematics

Given X1, X2, ... an infinite sequence of i.i.d. random variables with finite expected value E(X1) = E(X2) = ... = µ < ∞, we are interested in the convergence of the sample average

The weak lawEdit



This proof uses the assumption of finite variance   (for all  ). The independence of the random variables implies no correlation between them, and we have that


The common mean μ of the sequence is the mean of the sample average:


Using Chebyshev's inequality on   results in


This may be used to obtain the following:


As n approaches infinity, the expression approaches 1. And by definition of convergence in probability (see Convergence of random variables), we have obtained