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criterion for almost-sure convergence
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(Corollary)
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Let $X_1, X_2, \dotsc$ and $X$ be random variables. If, for every $\epsilon > 0$ the sum $\sum_{n=1}^\infty \PP( \abs{X_n - X} > \epsilon )$ is finite, then $X_n$ converge to $X$ almost surely.
Proof. By the Borel-Cantelli lemma, we have $\PP(\limsup_n \{ \abs{X_n - X} > \epsilon \})=0 $ But $\limsup_n \{ \abs{X_n - X} > \epsilon \}$ is the same as the event $\{ \limsup_n \abs{X_n - X} > \epsilon \}$ (The latter event involves the limit superior of numbers; the former involves the limit superior of sets.) So taking the limit $\epsilon \searrow 0$ we have $\PP( \limsup_n \abs{X_n - X} > 0) = 0$ or equivalently $\PP( \limsup_n \abs{X_n - X} = 0) = 1$ 
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"criterion for almost-sure convergence" is owned by stevecheng. [ full author list (2) | owner history (1) ]
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| Other names: |
corollary of Borel-Cantelli lemma |
This object's parent.
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Cross-references: limit, event, Borel-Cantelli lemma, almost surely, converge, finite, sum, random variables
This is version 12 of criterion for almost-sure convergence, born on 2006-05-18, modified 2006-10-22.
Object id is 7916, canonical name is CorollaryOfBorelCantelliLemma1.
Accessed 3782 times total.
Classification:
| AMS MSC: | 60A99 (Probability theory and stochastic processes :: Foundations of probability theory :: Miscellaneous) |
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Pending Errata and Addenda
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