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This is a problem that I am stuck at. I think I have to prove the hint first. But I can't find a way to prove the 'only if' part of the hint. (the 'if' part is just manifest). Could anyone help me how to prove the hint?

Also, I can't think of a way to prove that if $\frac{X_n}{n}$ goes to $0$ P-a.s, then $X_1$ is integrable when independence is assumed. I think I have to use some Kolmogorov's law of large numbers..but just stuck now. Please could anyone help me with this also?

Keith
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  • What does $n|\varepsilon$ mean? Is it just supposed to be multiplication $n\varepsilon$? It looks like a printing error. – Nate Eldredge May 04 '15 at 17:52
  • Oh, sorry. I changed the pictrue. Yes, it is multiplication. – Keith May 04 '15 at 17:55
  • Why does the step you say is "just manifest" seem manifest to you? – Did May 04 '15 at 17:58
  • This looks like a Borel-Cantelli type of problem (rather than Kolmogorov LLN). To prove the hint, you might use the fact that if $Y$ is a non-negative random variable, then $E[Y] = \int_0^{\infty} Pr[Y>y]dy$, and relate that integral to a sum. – Michael May 04 '15 at 17:58
  • Why does the equality hold? Could you prove it in more detail? – Keith May 04 '15 at 18:02
  • http://math.stackexchange.com/questions/64186/intuition-behind-using-complementary-cdf-to-compute-expectation-for-nonnegative – Michael May 04 '15 at 18:03
  • Thank you for your answer. Now, for proving "if (X_n)/n goes to 0 P-a.s, then X 1 is integrable" could you give me some advice? – Keith May 04 '15 at 18:05
  • Have you looked at Borel-Cantelli? – Michael May 04 '15 at 18:06
  • I am looking at it now, trying to figure out how to apply this lemma to the problem. But can't find a way yet. – Keith May 04 '15 at 18:11
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    I think I found a way out. Thank you for your help, – Keith May 04 '15 at 18:20

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