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$\{X_k\}_{k}$ are independent random variables on the probability space $(\Omega, \mathcal F, P)$ and $X_k$ has gamma density $f_k(x)$ where $f_k(x)=\dfrac{x^{a_x-1}e^{-x}}{\Gamma(a_k)}$ where $x,a_k >0$. Give necessary and sufficient conditions so $\sum_{k=1}^{\infty}X_k(\omega) < \infty$ almost surely.

I would like some hints on this I'm kind of stuck on this now. Here is what I have so far. Define $X(\omega):=\sum_{k=1}^{\infty}X_k(\omega)$. Since $X_k$ follows a gamma distribution then this means that $P(X_k<0) = 0$ so that $P(X_k \ge 0)=1$ then one sufficient condition would be that for every $\omega \in [X \ge 0],\;\;E(X)=E(\sum_{k=1}^{\infty}X_k)= \sum_{k=1}^{\infty}E(X_k) =\sum_{k=1}^{\infty}a_k < \infty$ because that would imply that $P(X=\infty)=0$ but I don't think that this condition is necessary.

Also it would be necessary that $X_k(\omega) \rightarrow 0$ as $k \rightarrow \infty$ but that is not sufficient.

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See Kolmogorov's "Three Series Theorem" if you want the definitive answer on this subject; it gives you results general for all distributions, not just the gamma distribution.

https://en.wikipedia.org/wiki/Kolmogorov%27s_three-series_theorem

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