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

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 subset of these random variables. For example, the event that the series

converges, where each X_i is nonnegative, 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, a sequence of 100 consecutive heads occurring 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.