Famous Theorems of Mathematics/Law of large numbers

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 law

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Theorem:  

Proof:

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