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I want to prove that for sequence of random variables $X_1,X_2,...$, that are independent either the sum $\sum_{i=1}^\infty X_i$ converges almost surely or diverges almost surely.

Any tips on how to prove this result? Particularly I have a problem as the definition for almost sure divergence is not provided.

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    $\begingroup$ Hewitt-Savage zero-one law $\endgroup$ – angryavian Oct 8 '18 at 22:25
  • $\begingroup$ @angryavian Thanks!! :) $\endgroup$ – Dole Oct 8 '18 at 22:30
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    $\begingroup$ @angryavian You need “iid” condition to apply Hewitt-Savage 0-1 law. The OP is not assuming this. $\endgroup$ – Rubertos Oct 8 '18 at 22:33
  • $\begingroup$ @Rubertos Good point, thanks for pointing out my error. $\endgroup$ – angryavian Oct 8 '18 at 22:59
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Note that if $\sum_{n=0}^\infty X_n(w)$ is convergent, then $\sum_{n=k}^\infty X_n(w)$ is convergent, and vice versa.

Now, apply Kolmogorov 0-1 law.

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