Kolmogorov's zero-one law

In probability theory, Kolmogorov's zero-one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, called a tail event, will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring is zero or one.

Tail events are defined in terms of infinite sequences of random variables. Suppose


 * $$X_1,X_2,X_3,\dots\,$$

is an infinite sequence of independent random variables (not necessarily identically distributed). Then, a tail event is an event whose occurrence or failure is determined by the values of these random variables but which is probabilistically independent of each finite subsequence of these random variables. For example, the event that the series


 * $$\sum_{k=1}^{\infty} X_k$$

converges, is a tail event. The event that the sum to which it converges is more than 1 is not a tail event, since, for example, it is not independent of the value of X1. In an infinite sequence of coin-tosses, the probability that a sequence of 100 consecutive heads occurs infinitely many times, is a tail event.

In many situations, it can be easy to apply Kolmogorov's zero-one law to show that some event has probability 0 or 1, but surprisingly hard to determine which of these two extreme values is the correct one.