# Show that for sequence of independent random variables $X_1,X_2,…$ $\sum_{i=1}^\infty X_i$ either converges or diverges almost surely

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.

• Hewitt-Savage zero-one law – angryavian Oct 8 '18 at 22:25
• @angryavian Thanks!! :) – Dole Oct 8 '18 at 22:30
• @angryavian You need “iid” condition to apply Hewitt-Savage 0-1 law. The OP is not assuming this. – Rubertos Oct 8 '18 at 22:33
• @Rubertos Good point, thanks for pointing out my error. – angryavian Oct 8 '18 at 22:59

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.