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 &sigma;-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 &sigma;-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 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 $$k\in\mathbb N$$ 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 &sigma;-algebras. Let (&Omega;,F,P) be a probability space and let Fn be a sequence of &sigma;-algebras contained in F. Let
 * $$G_n=\sigma\bigg(\bigcup_{k=n}^\infty F_k\bigg)$$

be the smallest &sigma;-algebra containing Fn, Fn+1, .... The terminal &sigma;-algebra of the Fn 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 Fn 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 Fn to be the &sigma;-algebra generated by the random variable Xn. A tail event is then by definition an event which is measurable with respect to the &sigma;-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 &sigma;-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 $$\frac{1}{2}$$.