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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!

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    $\begingroup$ What is the definition of uniform integrability that you were given? There are usually two parts in it, which part are you unable to show? (That you flout the rules of the site so squarely after 78 questions asked is worrisome.) $\endgroup$ – Did May 6 '13 at 6:10
  • $\begingroup$ I am not flouting any rule of the site. I dont understand why you keep commenting on my questions with no help. I dont want and need your comments. Please stop commenting on my problems. I am tired of reporting you for your unnecessary comments. $\endgroup$ – Salih Ucan May 7 '13 at 6:34
  • $\begingroup$ Your beliefs about the ways the site works are unfounded. (1) That a user comments on your questions, and what they comment, is not for you to decide. (2) That you are flouting the rules of the site is a fact (and what is so difficult about adding your thoughts on the questions you asked if you tried, even minimally, to solve them? even more troublesome, why do you fail to even ackowledge the queries for explanations made in comments?). (3) To signal this behaviour to other users, especially to newcomers, is useful unless they think the site is supposed to function the way you are using it. $\endgroup$ – Did May 7 '13 at 6:46
  • $\begingroup$ If you click on the flag next to comments you see that the question owner has the right to comment on what the others comment on his/her question. You better check the rules of this site more carefully. Moreover, you are not the one who decides what and how to post on this site. I am sorry that I am in this debate with you. All people who are posting questions are trying to do math and learn math unless some "quick homework answers" are asked which are obvious from the questions anyway. $\endgroup$ – Salih Ucan May 7 '13 at 6:55
  • $\begingroup$ Of course the question owner has the right to comment on what the others comment on their questions, did I ever write otherwise? (Feels like having to explain that $P\implies Q$ is not the same as $Q\implies P$...) // All people who are posting questions are trying to do math... Precisely, you are not. Copying on the site 78 verbatim questions you were asked to solve and waiting for the answers to appear without interacting with the answerers (even to correct obvious misprints when asked about them) is not, I repeat, is not, "doing maths". $\endgroup$ – Did May 7 '13 at 7:04
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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 $X_n:=n\chi_{(0,n^{-1})}$.

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