structural:scholar_edition
Proof of the weak law

Given X, X,... an infinite sequence of i.i.d. random variables with finite expected value E (X 1) = E (X 2) = ⋯ = μ < ∞ {\displaystyle E(X_{1})=E(X_{2})=\cdots =\mu <\infty }, we are interested in the convergence of the sample average

X ¯ n = 1 n (X 1 + ⋯ + X n). {\displaystyle {\overline {X}}_{n}={\tfrac {1}{n}}(X_{1}+\cdots +X_{n}).}

The weak law of large numbers states: