# Convergence in Probability to infinity

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? – user31373 Jul 4 '12 at 20:52
I don't see how this could help exactly. – Lenava Jul 6 '12 at 11:20
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 $|S_n|\to\infty$. – user31373 Jul 6 '12 at 16:04
Thank you. I will think about it and reply afterwards, later the day. – Lenava Jul 9 '12 at 7:16