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Let $\{X_n\}_{n \geq 1}$ be a sequence of independent random variables with $X_n \sim N(\mu_n, \sigma_n^2)$ for each $n \geq 1$. Show that $S_n = \sum_{i = 1}^n X_i$ converges almost surely if and only if both $\sum_{n = 1}^\infty \mu_n$ and $\sum_{n = 1}^\infty \sigma_n^2$ converge.

This has been answered here. The direction $\sum_{n = 1}^\infty \mu_n < \infty$ and $\sum_{n = 1}^\infty \sigma^2 < \infty$ implying $S_n$ converges almost surely is clear. I would like to prove the other direction without using characteristic functions, as this problem has been assigned to me in a course where we haven't encountered them yet.

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Almost sure convergence implies convergence in distribution, so it suffices to show the $S_n$ do not converge in distribution. We then have

$$S_n \sim N\left(\sum_{i=1}^n \mu_i, \sum_{i=1}^n \sigma^2_i\right),$$

which converges in distribution if and only if both sums converge.

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  • $\begingroup$ Thanks for the answer (+1)! But, can you please elaborate a little more on the "converges in distribution if and only if both sums converge" part.... $\endgroup$ Feb 17, 2023 at 12:53
  • $\begingroup$ The $S_n$ converge in distribution iff their distribution functions $F_n \to F$ pointwise, which obviously occurs iff the sums converge. @SayanDutta $\endgroup$
    – Adam
    Feb 17, 2023 at 16:16

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