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i was lately reading the book of Kallenberg "foundations of modern probability". I have a problem with understanding one of this thoughts(p. 70, Theorem 4.17):

Let $\xi_1,\xi_2,\ldots$ be independent symmetric randvom variables.

If $\sum_{k=0}^{\infty} \xi_k^2=\infty$ a.s. then $|S_n|=|\sum_{k\le n} \xi_k|$ converges in probability to infinity, i.e. for all $K>0$ $$P(|S_n|>K)\to 1(n\to \infty).$$

How do I see this implication? There is no argumentation, so I think it is pretty easy to see. But well, I can't.

Thank yopu for your help!

Regards, Lenava

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    Maybe the author used a form of CLT for non-identically distributed variables, such as [Lyapunov's](http://en.wikipedia.org/wiki/Central_limit_theorem#Lyapunov_CLT)?2012-07-04
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    I don't see how this could help exactly.2012-07-06
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    If you knew that $S_n/\alpha_n$ converges to a normal distribution, where $\alpha_n\to \infty$, that would probably help to show that $|S_n|\to\infty$.2012-07-06
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    Thank you. I will think about it and reply afterwards, later the day.2012-07-09

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