criterion for almost-sure convergence
Let and be random variables.
If, for every , the sum is finite,
then converge to almost surely.
Proof.
By the Borel-Cantelli lemma, we have .
But is the same as the event .
(The latter event involves the limit superior of numbers (http://planetmath.org/LimitSuperior); the former involves the
limit superior of sets (http://planetmath.org/InfinitelyOften).)
So taking the limit ,
we have ,
or equivalently
.
∎
Title | criterion for almost-sure convergence |
---|---|
Canonical name | CriterionForAlmostsureConvergence |
Date of creation | 2013-03-22 15:54:45 |
Last modified on | 2013-03-22 15:54:45 |
Owner | stevecheng (10074) |
Last modified by | stevecheng (10074) |
Numerical id | 15 |
Author | stevecheng (10074) |
Entry type | Corollary |
Classification | msc 60A99 |
Synonym | corollary of Borel-Cantelli lemma |