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I am trying to solve the following excercise.

Let $\{X_i\}$ be a sequence of independent stable random variables, that is $X_i \sim S_\alpha (\sigma_i,\beta_i,\mu_i)$ with $0<\alpha \leq 2$. The series $\sum_{i=1}^\infty X_i$ converges a.s. if and only if $\sum_{i=1}^\infty \mu_i$ converges and $\sum_{i=1}^\infty \sigma_i^{\alpha} < \infty$.

I tried to solve with the Kolmogorov's three series theorem, but I am lost. I will appreciate any suggestions and helps. Thanks!

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    $\begingroup$ By replacing $X_i$ with $X_i-\mu_i$ you may as well assume $\mu_i=0$. Furthermore, if you can prove convergence in probability then that is enough because then you can just apply this result. To prove convergence in probability of the given series, just use stability of the $X_i$ to get the distribution of $\sum_1^n X_i$, then use the given condition on $\sigma_i$. Conversely... well, you can do it. Someone may please elaborate using the full CF argument. $\endgroup$ – Shalop Jun 11 '18 at 23:30

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