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 $$(X_n)^\infty_{n=1}$$ 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.