# 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

is an infinite sequence of independent random variables (not necessarily identically distributed). Let be the σ-algebra generated by the . Then, a **tail event** is an event which is probabilistically independent of each finite subset of these random variables. (Note: belonging to implies that membership in is uniquely determined by the values of the 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.

Intuitively, tail events are precisely those events whose occurrence can still be determined if an arbitrarily large but finite initial segment of the 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[edit]

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 mutually independent σ-algebras contained in *F*. Let

be the smallest σ-algebra containing *F*_{n}, *F*_{n+1}, …. Then Kolmogorov's zero–one law asserts that for any event

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 *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 intersection .

## Examples[edit]

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*.

## See also[edit]

## References[edit]

- Stroock, Daniel (1999).
*Probability theory: An analytic view*(revised ed.). Cambridge University Press. ISBN 978-0-521-66349-6.. - Brzezniak, Zdzislaw; Zastawniak, Thomasz (2000).
*Basic Stochastic Processes*. Springer. ISBN 3-540-76175-6. - Rosenthal, Jeffrey S. (2006).
*A first look at rigorous probability theory*. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd. p. 37. ISBN 978-981-270-371-2.

## External links[edit]

- The Legacy of Andrei Nikolaevich Kolmogorov Curriculum Vitae and Biography. Kolmogorov School. Ph.D. students and descendants of A. N. Kolmogorov. A. N. Kolmogorov works, books, papers, articles. Photographs and Portraits of A. N. Kolmogorov.