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, namely a tail event of independent σ-algebras, 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 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 ${\mathcal {F}}$ be the sigma-algebra generated jointly by all of the $X_{k}$ . Then, a tail event $F\in {\mathcal {F}}$ is an event which is probabilistically independent of each finite subset of these random variables. (Note: $F$ belonging to ${\mathcal {F}}$ 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 $X_{k}=X_{k+1}=\dots =X_{k+100}=1$ 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 _{k=n}^{\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 ${\mathcal {T}}((F_{n})_{n\in \mathbb {N} })=\bigcap _{n=1}^{\infty }G_{n}$ .

Kolmogorov's zero–one law asserts that, if the F_n are stochastically independent, then for any event $E\in {\mathcal {T}}((F_{n})_{n\in \mathbb {N} })$ , 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 $\textstyle {\bigcap _{n=1}^{\infty }G_{n}}$ .

Examples

An invertible measure-preserving transformation on a standard probability space that obeys the 0-1 law is called a Kolmogorov automorphism.^{[clarification needed]} All Bernoulli automorphisms 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.

Let $\{X_{n}\}_{n}$ be a sequence of independent random variables, then the event $\left\{\lim _{n\rightarrow \infty }\sum _{k=1}^{n}X_{k}{\text{ exists }}\right\}$ 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 ${\frac {1}{2}}$ each. In this case the sum converges with probability ${\frac {1}{2}}$ .

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.

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