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, 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 for . Let be the sigma-algebra generated jointly by all of the . Then, a tail event is an event which is probabilistically independent of each finite subset of these random variables. (Note: belonging to implies that membership in is uniquely determined by the values of the , 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 converges, and the event that its sum converges are both tail events. If the are, for example, all Bernoulli-distributed, then the event that there are infinitely many such that is a tail event. If each models the outcome of the -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 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. 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 .