Kolmogorov's zero–one law

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"Tail event" redirects here. For "tail events" meaning "rare events", see fat tail.

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.

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. All Bernoulli automorphisms are Kolmogorov automorphisms but not vice versa.