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Example of non-constant random variables $X_n$ such that $\sum_{i=1}^n X_i^2$ has exponential growth a.s. as $n$ goes to infinity (for instance, $\sum_{i=1}^n X_i^2 \sim \exp(n)$ a.s.).

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Let $X_1=\pm e^{1/2}$ (the two possibilities being equally likely) and $X_i=\pm \sqrt{e^i-e^{i-1}}$ for $i>1$. Then $\sum_1^n X_i^2=e^n$ surely.

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Thanks but my purpuse is to get random variables X_n dpending on omega. – Andrew Matthew Dec 31 '12 at 10:59
@AndrewMatthew: Then what's the problem? These depend on omega: none of them are constant. – Chris Eagle Dec 31 '12 at 12:05

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