# 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 $X_{k}$ for $k\in \mathbb {N}$. Let ${\mathcal {F}}$ be the sigma-algebra generated jointly by all of the $X_{k}$. Then, a tail event $F\in {\mathcal {F}}$ is an event the occurrence of which cannot depend on the outcome of a finite subfamily 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 $k\in \mathbb {N}$ such that $X_{k}=X_{k+1}=\dots =X_{k+100}=1$ 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 F$_n$ be a sequence of σ-algebras contained in F. Let
    $G_{n}=\sigma {\bigg\bigcup _{k=n}^{\infty }F_{k}{\bigg )}$
be the smallest σ-algebra containing F$_n$, F$_{n+1}$, .... The terminal σ-algebra of the F$_n$ is defined as ${\mathcal {T}}((F_{n})_{n\in \mathbb {N} })=\bigcap _{n=1}^{\infty }G_{n}$.
Kolmogorov's zero–one law asserts that, if the F$_n$ are stochastically independent, then for any event $E\in {\mathcal {T}}((F_{n})_{n\in \mathbb {N} })$, 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 F$_n$ to be the σ-algebra generated by the random variable X$_n$. A tail event is then by definition an event which is measurable with respect to the σ-algebra generated by all X$_n$, but which is independent of any finite number of X$_n$. That is, a tail event is precisely an element of the terminal σ-algebra $\textstyle {\bigcap _{n=1}^{\infty }G_{n}}$.

## Examples

An invertible measure-preserving transformation on a standard probability space that obeys the 0-1 law is called a Kolmogorov automorphism. 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 $\left\{\lim _{n\rightarrow \infty }\sum _{k=1}^{n}X_{k}{\text{ exists }}\right\}$ is a tail event. Thus by Kolmogorov 0-1 law, it has either probability 0 or 1 to happen. Note that independence is required for the tail event condition to hold. Without independence we can consider a sequence that's either $(0,0,0,\dots )$ or $(1,1,1,\dots )$ with probability ${\frac {1}{2}}$ each. In this case the sum converges with probability ${\frac {1}{2}}$.