Kolmogorov's zero–one law
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In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, Kolmogorov's zero–one law, named in honor of
Andrey Nikolaevich Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
, specifies that a certain type of
event Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of ev ...
, namely a ''tail event of independent σ-algebras'', will either
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
happen or almost surely not happen; that is, the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
of such an event occurring is zero or one. 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 X_k for k\in\mathbb N. Let \mathcal be the sigma-algebra generated jointly by all of the X_k. Then, a tail event F \in \mathcal is an event which is probabilistically independent of each finite subset of these random variables. (Note: F belonging to \mathcal 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 probability that there are infinitely many k\in\mathbb N such that X_k=X_=\dots=X_=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 σ-algebras. Let (Ω,''F'',''P'') be a probability space and let ''F''''n'' be a sequence of σ-algebras contained in ''F''. Let :G_n=\sigma\bigg(\bigcup_^\infty F_k\bigg) be the smallest σ-algebra containing ''F''''n'', ''F''''n''+1, …. The ''terminal σ-algebra'' of the ''F''''n'' is defined as \mathcal T((F_n)_)=\bigcap_^\infty G_n. Kolmogorov's zero–one law asserts that, if the ''F''''n'' are stochastically independent, then for any event E\in \mathcal T((F_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 ''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 terminal σ-algebra \textstyle.


Examples

An invertible
measure-preserving transformation In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special ca ...
on a
standard probability space In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin ...
that obeys the 0-1 law is called a
Kolmogorov automorphism In mathematics, a Kolmogorov automorphism, ''K''-automorphism, ''K''-shift or ''K''-system is an invertible, measure-preserving automorphism defined on a standard probability space that obeys Kolmogorov's zero–one law.Peter Walters, ''An Introdu ...
. All
Bernoulli automorphism In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical sys ...
s 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.


See also

* Borel–Cantelli lemma *
Hewitt–Savage zero–one law The Hewitt–Savage zero–one law is a theorem in probability theory, similar to Kolmogorov's zero–one law and the Borel–Cantelli lemma, that specifies that a certain type of event will either almost surely happen or almost surely not happen. I ...
*
Lévy's zero–one law In mathematicsspecifically, in the theory of stochastic processesDoob's martingale convergence theorems are a collection of results on the limits of supermartingales, named after the American mathematician Joseph L. Doob. Informally, the marting ...
* Long tail * Tail risk


References

*. * *


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. {{DEFAULTSORT:Kolmogorov's zero-one law Probability theorems Covering lemmas