Theorem

corollaries of Kolmogorov zero-one law

Let $\{X_n\}$ be independent random variables. Then the following are true.

1. The event $[\sum_{n}X_n \text{ converges}]$ has probability 0 or 1.
2. The random variables $\limsup_{n \to \infty}X_n$ and $\liminf_{n \to \infty}X_n$ are constant with probability 1.
3. The event $\{\omega: S_{n}(\omega)/n \to 0\}$ has probability 0 or 1. (We define $S_n \overset{\Delta}{=} X_1 + \cdots + X_n$.)