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 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