Definition:Almost Sure Convergence
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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 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:
- $X_n \xrightarrow {\text {a.s.}} X$
Sources
- 2002: George C. Casella and Roger L. Berger: Statistical Inference (2nd ed.): $5.5$: Convergence Concepts: Definition $5.5.6$