Under what conditions $E|X_n-X|\rightarrow 0$, given $X_n\rightarrow X$?

Question

Given we have a sequence of random variables $X_n\rightarrow X$, under what conditions can we say that $E|X_n-X|\rightarrow 0$?

I know that $X_n\rightarrow X$ implies that $|X_n-X|\rightarrow 0$, but this alone is not enough. The dominated convergence theorem for expectation implies the desired result but that assumes that $|X_n|\leq Z\in L^1$.

Also if $\|X_n-X\|_\infty \rightarrow 0$, that should do it.

Is there a necessary condition?

Answer 1

If the family $X_n$ is equi-integrable and $X_n$ converges to $X$ in probability then:

• $X$ is integrable and $E|X_n-X| \rightarrow 0$.

Equi-integrable means that the function: $$ M(R) = \sup_n E \left( |X_n| 1_{|X_n|>R} \right) = \sup_n \int_{|X_n|>R} |X_n| \; dP$$ goes to zero as $R\rightarrow \infty$.