# 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 ${\displaystyle X_{k}}$ for ${\displaystyle k\in \mathbb {N} }$. Let ${\displaystyle {\mathcal {F}}}$ be the sigma-algebra generated jointly by all of the ${\displaystyle X_{k}}$. Then, a tail event ${\displaystyle F\in {\mathcal {F}}}$ is an event which is probabilistically independent of each finite subset of these random variables. (Note: ${\displaystyle F}$ belonging to ${\displaystyle {\mathcal {F}}}$ implies that membership in ${\displaystyle F}$ is uniquely determined by the values of the ${\displaystyle X_{k}}$, 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 ${\displaystyle X_{k}}$ converges, and the event that its sum converges are both tail events. If the ${\displaystyle X_{k}}$ are, for example, all Bernoulli-distributed, then the event that there are infinitely many ${\displaystyle k\in \mathbb {N} }$ such that ${\displaystyle X_{k}=X_{k+1}=\dots =X_{k+100}=1}$ is a tail event. If each ${\displaystyle X_{k}}$ models the outcome of the ${\displaystyle k}$-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 ${\displaystyle X_{k}}$ 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

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

be the smallest σ-algebra containing Fn, Fn+1, .... The terminal σ-algebra of the Fn is defined as ${\displaystyle {\mathcal {T}}((F_{n})_{n\in \mathbb {N} })=\bigcap _{n=1}^{\infty }G_{n}}$.

Kolmogorov's zero–one law asserts that, if the Fn are stochastically independent, then for any event ${\displaystyle E\in {\mathcal {T}}((F_{n})_{n\in \mathbb {N} })}$, 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 ${\displaystyle \textstyle {\bigcap _{n=1}^{\infty }G_{n}}}$.

## 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. The presence of an infinite cluster in the context of percolation theory also obeys the 0-1 law.

Let ${\displaystyle \{X_{n}\}_{n}}$ be a sequence of random variable, then the event ${\displaystyle \left\{\lim _{n\rightarrow \infty }\sum _{k=1}^{n}X_{k}{\text{ exists }}\right\}}$ is a tail event. Thus by Kolmogorov 0-1 law, it has either probability 0 or 1 to happen.