# 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). Let ${\mathcal {F}}$ be the σ-algebra generated by the $X_{i}$ . Then, a tail event $F\in {\mathcal {F}}$ 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 $X_{i}$ but the latter condition is strictly weaker and does not suffice to prove the zero-one law.) For example, the event that the sequence converges, and the event that its sum converges are both tail events. In an infinite sequence of coin-tosses, a sequence of 100 consecutive heads occurring infinitely many times is a tail event.

Tail events are precisely those events whose occurrence can still be determined if an arbitrarily large but finite initial segment of the $X_{i}$ are 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 mutually independent σ-algebras contained in F. Let

$G_{n}=\sigma {\bigg (}\bigcup _{k=n}^{\infty }F_{k}{\bigg )}$

be the smallest σ-algebra containing Fn, Fn+1, …. Then Kolmogorov's zero–one law asserts that for any event

$F\in \bigcap _{n=1}^{\infty }G_{n}$

one has either P(F) = 0 or 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 intersection $\textstyle {\bigcap _{n=1}^{\infty }G_{n}}$ .

## Examples

An invertible measure-preserving transformation on a standard probability space that obeys the 0-1 law is called a Kolmogorov automorphism.[clarification needed] All Bernoulli automorphisms are Kolmogorov automorphisms but not vice versa.