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
Xk
k\inN
l{F}
Xk
F\inl{F}
F
l{F}
F
Xk
Xk
Xk
k\inN
Xk=Xk+1=...=Xk+100=1
Xk
k
Tail events are precisely those events whose occurrence can still be determined if an arbitrarily large but finite initial segment of the
Xk
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.
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 σ-algebras contained in F. Let
Gn=\sigma(cup
infty | |
k=n |
Fk)
lT((Fn)n\inN
infty | |
)=cap | |
n=1 |
Gn
Kolmogorov's zero–one law asserts that, if the Fn are stochastically independent, then for any event
E\inlT((Fn)n\inN)
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 terminal σ-algebra
infty | |
style{cap | |
n=1 |
Gn}
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. The presence of an infinite cluster in the context of percolation theory also obeys the 0-1 law.
Let
\{Xn\}n
\left\{\limn
n | |
\sum | |
k=1 |
Xkexists\right\}
(0,0,0,...)
(1,1,1,...)
1 | |
2 |
1 | |
2 |