In probability theory, Kolmogorov's zero–one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, namely a tail event of independent σ-algebras, 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 countably infinite families of σ-algebras. For illustrative purposes, we present here the special case in which each sigma algebra is generated by a random variable X_k for. Let \mathcal{F} be the sigma-algebra generated jointly by all of the X_k. Then, a tail event is an event which is probabilistically independent of each finite subset of these random variables. (Note: F belonging to \mathcal{F} implies that membership in F is uniquely determined by the values of the X_k, but the latter condition is strictly weaker and does not suffice to prove the zero-one law.) For example, the event that the sequence of the X_k converges, and the event that its sum converges are both tail events. If the X_k are, for example, all Bernoulli-distributed, then the event that there are infinitely many such that is a tail event. If each X_k models the outcome of the k-th coin toss in a modeled, infinite sequence of coin tosses, this means that a sequence of 100 consecutive heads occurring infinitely many times is a tail event in this model. Tail events are precisely those events whose occurrence can still be determined if an arbitrarily large but finite initial segment of the X_k is removed. 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.

Formulation

A more general statement of Kolmogorov's zero–one law holds for sequences of independent σ-algebras. Let (Ω,F,P) be a probability space and let Fn be a sequence of σ-algebras contained in F. Let be the smallest σ-algebra containing Fn, Fn+1, .... The terminal σ-algebra of the Fn is defined as. Kolmogorov's zero–one law asserts that, if the Fn are stochastically independent, then for any event, one has either P(E) = 0 or P(E)=1. The statement of the law in terms of random variables is obtained from the latter by taking each Fn to be the σ-algebra generated by the random variable Xn. A tail event is then by definition an event which is measurable with respect to the σ-algebra generated by all Xn, but which is independent of any finite number of Xn. That is, a tail event is precisely an element of the terminal σ-algebra.

Examples

An invertible measure-preserving transformation on a standard probability space that obeys the 0-1 law is called a Kolmogorov automorphism. All Bernoulli automorphisms are Kolmogorov automorphisms but not vice versa. The presence of an infinite cluster in the context of percolation theory also obeys the 0-1 law. Let {X_n}_n be a sequence of independent random variables, then the event is a tail event. Thus by Kolmogorov 0-1 law, it has either probability 0 or 1 to happen. Note that independence is required for the tail event condition to hold. Without independence we can consider a sequence that's either or with probability \frac{1}{2} each. In this case the sum converges with probability \frac{1}{2}.