Kolmogorov's zero–one law

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"Tail event" redirects here. For "tail events" meaning "rare events", see fat tail.

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

X_1,X_2,X_3,\dots\,

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 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.

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 Fn be a sequence of mutually independent σ-algebras contained in F. Let

G_n=\sigma\bigg(\bigcup_{k=n}^\infty F_k\bigg)

be the smallest σ-algebra containing Fn, Fn+1, …. Then Kolmogorov's zero–one law asserts that for any event

F\in \bigcap_{n=1}^\infty G_n

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 \textstyle{\bigcap_{n=1}^\infty G_n}.

Examples[edit]

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[edit]

References[edit]

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.