Almost Sure Convergence

A sequence of random variables, $X_1, X_2,\ldots$ converges almost surely to a random variable $X$ if, for every $\varepsilon > 0,$ \[P(\lim_{n\rightarrow\infty}|X_n-X|<\varepsilon) = 1.\]

Theorem 5.6.1 (Strong Law of Large Numbers) Let $X_1,X_2,\ldots$ be iid random variables weith $EX_i=\mu$ and $Var\,X_i=\sigma^2 < \infty,$ and define $\overline{X}_n=(1/n)\sum_{i=1}^n X_i.$ Then, for every $\varepsilon>0,$ \[P(\lim_{n\rightarrow\infty}|\overline{X}_n-\mu|<\varepsilon) = 1\] That is, $\overline{X}_n$ converges almost surely to $\mu.$