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Suppose $\operatorname{Var}(X_i) = \sigma^2_i$ and let $S_n = \sum_{i=1}^n \sigma^2_i = O(1)$, then I need to find $\lim_{n \to \infty} S_n$. Do we have $S_n \to \infty$?

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  • $\begingroup$ Hello! You may find it helpful to look at the Mathjax formatting guide to learn how to write mathematical symbols here. $\endgroup$ – user296602 Feb 3 '16 at 20:31
  • $\begingroup$ I made some edits, trying to minimize the wording edits. $\endgroup$ – Ian Feb 3 '16 at 20:33
  • $\begingroup$ Also, this is not really tractable; all your hypothesis implies is that $\lim_{n \to \infty} S_n$ is a finite number given by $\sum_{i=1}^\infty \sigma_i^2$. But this limit could potentially be any nonnegative number. $\endgroup$ – Ian Feb 3 '16 at 20:34
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If $\sigma(i)={1\over i}$, $S_n=O(1)$ but $\lim_{n\to\infty} S_n={\pi^2\over6}$. So the limit may not be $+\infty$.

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