# Sums of random variables with exponential growth

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.