Definition:Almost Sure Convergence

Definition
Let $\sequence {X_n}_{n \ge 1}$ be a sequence of random variables.

Let $X$ be a random variable.

We say that $\sequence {X_n}$ almost surely converges to $X$ if:


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

for all real $\varepsilon > 0$.

This is written:


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