uniform integrability characterization

Question

How to show the following:

When a family of random variables $ \{X_n\}_{n \geq 1}$ is $L^p$ bounded for some $p > 1$ then $ \{X_n\}_{n \geq 1}$ is uniformly integrable.

Also why does the above statement fail for $p \leq 1$? Could you give counterexamples?

Thanks a lot!

Answer 1

Hint: whatever definition of uniform integrability you use, you can tackle the problem using Hölder's inequality. De la Vallée-Poussin's theorem gives a partial converse.

A counter-example could be $X_n:=n\chi_{(0,n^{-1})}$.