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This is a problem after the section "Strong Law of Large Numbers" of Shiryaev's Probability:

Let $\xi_1,\xi_2,...$ denote independent and identically distributed random variables such thatt $E|\xi_1|=\infty$. Show that $$\limsup_{n\to\infty}\left|\frac{S_n}{n}-a_n\right|=\infty\text{ (P-a.s.)}$$ for every sequence of constants $\{a_n\}$.

I have no idea about it. Any hint please. Thanks!

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1. as a tail event it occurs with probability 0 or 1, 2. symmetrize $X_i \rightarrow X_i - \tilde X_i$ so the $X_i$ are symmetric and $a_i = 0$. ${ }{ }$ 3. show can't be $ < k$ on $X_i > ik$ i.o., which occurs by lack of first moment –  mike Jan 10 '13 at 14:36
@mike Thanks! But I can't catch the hint 2 clearly. Does it meam let $Z_i=X_i-Y_i$ where $Y_i$ are also iid?If so how can I prove $Z_i$ lack of first moment? –  Danielsen Jan 10 '13 at 15:47
yes, and lack of first moment is a standard application of fubini: $\infty > \mathbb E(|X - Y|) = \int \int (|x-y| \mu(dx)\mu(dy) = \int \mathbb E(|X - y|) \mu(dy) $ which implies that $\infty > \mathbb E(|X - y|) $ a.e. y which implies $\infty > \mathbb E(|X |)$ –  mike Jan 10 '13 at 15:55
@mike I get it. Thank you very much! –  Danielsen Jan 10 '13 at 16:02
@Danielsen You might wish to write down a full solution based on mike's hints, to post it here as an answer and even, after a while and if no other solution appears that you would prefer, to accept it. –  Did Jan 11 '13 at 18:20

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