Lévy's convergence theorem
- Lévy's convergence theorem
In probability theory Lévy's convergence theorem (sometimes also called Lévy's dominated convergence theorem) states that for a sequence of random variables where
* and
* where "Y" is some random variable with
*
it follows that
*
*
*.
Essentially, it is a sufficient condition for the almost sure convergence to imply "L"1-convergence.The condition could be relaxed. Instead, the sequence should be uniformly integrable.
The theorem is simply a special case of Lebesgue's dominated convergence theorem in measure theory.
ee also
* Convergence of random variables
* Fatou's lemma
References
*A.N.Shiryaev (1995). "Probability, 2nd Edition", Springer-Verlag, New York, pp.187-188, ISBN 978-0387945491
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