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 $(X\_n)^\backslash infty\_\{n=1\}$ where

• $X\_n\backslash xrightarrow\{a.s.\}\; X$ and
• where Y is some random variable with

it follows that

• $\backslash mathrm\{E\}X\_n\backslash to\; \backslash mathrm\{E\}\; X$
• $\backslash mathrm\{E\}\; |X-X\_n|\backslash to\; 0$.

Essentially, it is a sufficient condition for the almost sure convergence to imply L¹-convergence. The condition could be relaxed. Instead, the sequence $(X\_n)^\backslash infty\_\{n=1\}$ should be uniformly integrable.

The theorem is simply a special case of Lebesgue\'s dominated convergence theorem in measure theory.

See 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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