Definition:Almost Sure Convergence

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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 and only if:

$\forall \epsilon \in \R_{>0}: \ds \map \Pr {\lim_{n \mathop \to \infty} \size {X_n - X} < \epsilon} = 1$

This is written:

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