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.

Tail events are defined in terms of infinite sequences of random variables. Suppose

${\displaystyle X_{1},X_{2},X_{3},\dots }$

is an infinite sequence of independent random variables (not necessarily identically distributed). Let ${\displaystyle {\mathcal {F}}}$ be the σ-algebra generated by the ${\displaystyle X_{i}}$. Then, a tail event ${\displaystyle F\in {\mathcal {F}}}$ is an event which is probabilistically independent of each finite subset of these random variables. (Note: ${\displaystyle F}$ belonging to ${\displaystyle {\mathcal {F}}}$ implies that membership in ${\displaystyle F}$ is uniquely determined by the values of the ${\displaystyle X_{i}}$ but the latter condition is strictly weaker and does not suffice to prove the zero-one law.) 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.

Tail events are precisely those events whose occurrence can still be determined if an arbitrarily large but finite initial segment of the ${\displaystyle X_{i}}$ are 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 mutually independent σ-algebras contained in F. Let

${\displaystyle G_{n}=\sigma {\bigg (}\bigcup _{k=n}^{\infty }F_{k}{\bigg )}}$

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

${\displaystyle F\in \bigcap _{n=1}^{\infty }G_{n}}$

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 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 intersection ${\displaystyle \textstyle {\bigcap _{n=1}^{\infty }G_{n}}}$.

Examples

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

• Borel–Cantelli lemma
• Hewitt–Savage zero–one law
• Lévy's zero–one law
• Long tail
• Tail risk

References

• Stroock, Daniel (1999). Probability theory: An analytic view (revised ed.). Cambridge University Press. ISBN 978-0-521-66349-6..
• Brzezniak, Zdzislaw; Zastawniak, Thomasz (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.

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.