Kolmogorov's zero–one law
In
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 for . Let be the sigma-algebra generated jointly by all of the . Then, a tail event is an event which is
Tail events are precisely those events whose occurrence can still be determined if an arbitrarily large but finite initial segment of the 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 Fn be a sequence of σ-algebras contained in F. Let
be the smallest σ-algebra containing Fn, Fn+1, .... The terminal σ-algebra of the Fn is defined as .
Kolmogorov's zero–one law asserts that, if the Fn are stochastically independent, then for any event , 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 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 terminal σ-algebra .
Examples
An
Let be a sequence of independent random variables, then the event 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 or with probability each. In this case the sum converges with probability .
See also
- Borel–Cantelli lemma
- Hewitt–Savage zero–one law
- Lévy's zero–one law
- Tail sigma-algebra
- Long tail
- Tail risk
References
- Stroock, Daniel (1999). Probability theory: An analytic view (revised ed.). ISBN 978-0-521-66349-6..
- Brzezniak, Zdzislaw; Zastawniak, Thomasz (2000). Basic Stochastic Processes. 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
- 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.