criterion for almost-sure convergence

Let $X_{1},X_{2},\\ldots$ and $X$ be random variables.If, for every $\\epsilon>0$ , the sum $\\sum_{n=1}^{\\infty}\\mathbb{P}(\\lvert X_{n}-X\\rvert>\\epsilon)$ is finite,then $X_{n}$ converge to $X$ almost surely.

Proof.

By the Borel-Cantelli lemma, we have $\\mathbb{P}(\\limsup_{n}\\{\\lvert X_{n}-X\\rvert>\\epsilon\\})=0$ .But $\\limsup_{n}\\{\\lvert X_{n}-X\\rvert>\\epsilon\\}$ is the same as the event $\\{\\limsup_{n}\\lvert X_{n}-X\\rvert>\\epsilon\\}$ .(The latter event involves the limit superior of numbers (http://planetmath.org/LimitSuperior); the former involves thelimit superior of sets (http://planetmath.org/InfinitelyOften).)So taking the limit $\\epsilon\\searrow 0$ ,we have $\\mathbb{P}(\\limsup_{n}\\lvert X_{n}-X\\rvert>0)=0$ ,or equivalently $\\mathbb{P}(\\limsup_{n}\\lvert X_{n}-X\\rvert=0)=1$ .∎

单词	CriterionForAlmostsureConvergence
释义	criterion for almost-sure convergence Let $X_{1},X_{2},\\ldots$ and $X$ be random variables.If, for every $\\epsilon>0$ , the sum $\\sum_{n=1}^{\\infty}\\mathbb{P}(\\lvert X_{n}-X\\rvert>\\epsilon)$ is finite,then $X_{n}$ converge to $X$ almost surely. Proof. By the Borel-Cantelli lemma, we have $\\mathbb{P}(\\limsup_{n}\\{\\lvert X_{n}-X\\rvert>\\epsilon\\})=0$ .But $\\limsup_{n}\\{\\lvert X_{n}-X\\rvert>\\epsilon\\}$ is the same as the event $\\{\\limsup_{n}\\lvert X_{n}-X\\rvert>\\epsilon\\}$ .(The latter event involves the limit superior of numbers (http://planetmath.org/LimitSuperior); the former involves thelimit superior of sets (http://planetmath.org/InfinitelyOften).)So taking the limit $\\epsilon\\searrow 0$ ,we have $\\mathbb{P}(\\limsup_{n}\\lvert X_{n}-X\\rvert>0)=0$ ,or equivalently $\\mathbb{P}(\\limsup_{n}\\lvert X_{n}-X\\rvert=0)=1$ .∎
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