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.
is an infinite sequence of independent random variables (not necessarily identically distributed). Then, a tail event is an event whose occurrence or failure is determined by the values of these random variables but which is probabilistically independent of each finite subset of these random variables. For example, the event that the series
converges, where each Xi is nonnegative, is a tail event. The event that the sum to which it converges is more than 1 is not a tail event, since, for example, it is not independent of the value of X1. 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.
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
be the smallest σ-algebra containing Fn, Fn+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 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 .
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.
See also 
- Stroock, Daniel (1999), Probability theory: An analytic view (revised ed.), Cambridge University Press, ISBN 978-0-521-66349-6.
- Brzezniak, Zdzislaw; Tomasz Zastawniak (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
- 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.