Lévy's convergence theorem

From Wikipedia, the free encyclopedia

In probability theory Lévy's convergence theorem (sometimes also called Lévy's dominated convergence theorem) states that for a sequence of random variables $(X_n)^\infty_{n=1}$ where

$X_n\xrightarrow{a.s.} X$ and
$| X n | < Y,$ where Y is some random variable with
$\mathrm{E}Y < \infty$

it follows that

$\mathrm{E}|X| < \infty,$
$\mathrm{E}X_n\to \mathrm{E} X$
$\mathrm{E} |X-X_n|\to 0$ .

Essentially, it is a sufficient condition for the almost sure convergence to imply L¹-convergence. The condition $|X_n| < Y,\; \mathrm{E}Y < \infty$ could be relaxed. Instead, the sequence $(X_n)^\infty_{n=1}$ should be uniformly integrable.