Kolmogorov's Zero–one Law
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In
probability theory Probability theory or probability calculus 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 expre ...
, Kolmogorov's zero–one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, 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 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur ha ...
happen or almost surely not happen; that is, the
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
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 the occurrence of which cannot depend on the outcome of a finite subfamily 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 event 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 In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models ...
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 In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
measure-preserving transformation on a
standard probability space Standard may refer to: Symbols * Colours, standards and guidons, kinds of military signs * Standard (emblem), a type of a large symbol or emblem used for identification Norms, conventions or requirements * Standard (metrology), an object t ...
that obeys the 0-1 law is called a Kolmogorov automorphism. All Bernoulli automorphisms are Kolmogorov automorphisms but not ''vice versa''. The presence of an infinite cluster in the context of
percolation theory In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected ...
also obeys the 0-1 law. Let \_n be a sequence of independent random variables, then the event \left\ 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 (0,0,0,\dots) or (1,1,1,\dots) with probability \frac each. In this case the sum converges with probability \frac.


See also

*
Borel–Cantelli lemma In probability theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli, who gave statement to the lemma in the first d ...
* Hewitt–Savage zero–one law *
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 martin ...
* Tail sigma-algebra *
Long tail In statistics and business, a long tail of some distributions of numbers is the portion of the distribution having many occurrences far from the "head" or central part of the distribution. The distribution could involve popularities, random n ...
* Tail risk


References

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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 Theorems in probability theory Covering lemmas