When does mean convergence implies almost sure convergence?

Question

Suppose I have a sequence $(X_n)$ of random variables such that $X_n\to X$ in mean $(L_1)$.

What are the conditions that I need to impose so that I can affirm that $X_n\to X$ almost surely?

I know that Lebesgue's Dominated Convergence Theorem can be utilized to show that a sequence of random variables that converges almost surely converges in mean. But for this I do not know any conditions.

Answer 1

One simple sufficient condition is $\sum E|X_n -X| <\infty$. This condition implies $\sum |X_n -X| <\infty$ almost surely,which implies $X_n \to X$ almost surely.