# Almost Sure maximum zero [closed]

Given $E(|X_1|)$ is finite, let $X_n$ be r.v's identically distribued but not necessarily independent. Show that

$$\lim_{n\to +\infty} \frac 1n \max_{1\leq j\leq n} |X_j| =0 \mbox{ a.s.}$$

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Can you tell us what you have tried and where you are stuck? –  Alex Becker Sep 9 '12 at 4:08
(1\n) *(max |X_n|) \leq (1 \ n) * \sum_1^n |X_i| –  Salih Ucan Sep 9 '12 at 4:36
so it is integrable definitely then?? no LDT fatou or monotone??? –  Salih Ucan Sep 9 '12 at 4:36
Hint: consider truncated random variables $Y_n=X_n$ if $|X_n|\ge M$ and $Y_n=0$ otherwise. Apply the estimate in your comment to $Y_n$ instead of $X_n$. It will work better because the truncated variables have small first moment. Finally, observe that the maximum of $X_n$ can't be large without the maximum of $Y_n$ also being large.