Definition:Almost Sure Convergence

Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\sequence {X_n}_{n \mathop \in \N}$ be a sequence of real-valued random variables on $\struct {\Omega, \Sigma, \Pr}$.

Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.

We say that $\sequence {X_n}_{n \mathop \in \N}$ almost surely converges to $X$ if:


 * $\ds \map \Pr {\lim_{n \mathop \to \infty} \size {X_n - X} < \epsilon} = 1$

for all real $\epsilon > 0$.

This is written:


 * $X_n \xrightarrow {\text {a.s.}} X$