Convergence into law of a random series

Question

Theorem: Let $(X_n)$ be a sequence of real random variables independent on $(\Omega; F; P)$. The following assertions are equivalent.

1. $\sum X_n$ converges to the law.
2. $\sum X_n$ converges in probability.
3. $\sum X_n$ converges almost surely.

Where can I find the proof of this theorem? Thank you.

Answer 1

Please refer to the following books:
J. Neveu, Mathematical Foundations of the Calculus of Probability, Holden-Day, Inc. 1965. p.157, Corollary IV.7.1.
S. I. Resnick, A Probability Path, Birkhäuser Boston, 2005. p.211, Th 7.3.2.