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I want to show the Strong law of large numbers, That is, $$$$

$\frac{X_1+...+X_n}{n} \rightarrow 0$(almost surely) $$$$ where each ${X_i}$ are independent random variable with finite third moments with same density function.

First I want to show under hypothesis that second moments are finite, that $$$$$\sum_{n=1}^{\infty} nP(|X|>n)<\infty$ $$$$ But I don't know how to prove this statement. I tried to used weak theorem of large numbers but it seems like strong law can't be deduced from weak one. $$$$Also, if I prove this, how can I proceed to show the Strong law of large numbers? I'm stuck here.. Actually, if someone helped me out just for the first part, namely proving $\sum_{n=1}^{\infty} nP(|X|>n)<\infty$, I would really appreciate it.

StubbornAtom
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1 Answers1

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To prove that $\sum_{n=1}^{\infty} nP(|X|>n)<\infty$ when $E(X^2)<\infty$, you can use the following results:

  1. Proof that $E[X^2]$ = $\sum_{n=1}^\infty (2n-1) P(X\ge n)$

  2. Show $E[X] = \Sigma^\infty_{n=1}P[X\ge n]$

grand_chat
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