Kolmogorov's three-series theorem

Remarks

If the three series converge for some $A > 0$, then they converge for every $A > 0$. By Kolmogorov's zero–one law, the event that $\sum_n X_n$ converges has probability $0$ or $1$, so when the series do not all converge, $\sum_n X_n$ diverges almost surely.

Idea of the proof of sufficiency. Series 3 and Kolmogorov's inequality show that $\sum_n (Y_n - E[Y_n])$ converges almost surely. Series 2 then gives the convergence of $\sum_n Y_n$. Series 1 and the first Borel–Cantelli lemma show that $X_n = Y_n$ for all but finitely many $n$, almost surely. The proof of necessity uses a symmetrisation argument and a lower bound for the probability that a sum of bounded centred variables stays small.

Example: let $\varepsilon_n$ be independent signs, equal to $1$ or $-1$ with probability $1/2$ each, and let $X_n = \varepsilon_n / n^{p}$ with $p > 0$. With $A = 1$, series 1 and 2 are $0$, and series 3 is $\sum_n n^{-2p}$. So the random series $\sum_n \varepsilon_n / n^p$ converges almost surely if $p > 1/2$ and diverges almost surely if $p \le 1/2$.

Going deeper

For a student

The theorem completely describes the almost sure convergence of series of independent variables. It combines Kolmogorov's inequality, the Borel–Cantelli lemma and the zero–one law. For independent variables, almost sure convergence, convergence in probability and convergence in distribution of $\sum_n X_n$ are equivalent, a result of Lévy. A common use is to show that random series such as $\sum_n \varepsilon_n a_n$ converge almost surely exactly when $\sum_n a_n^2 < \infty$.