Kolmogorov's zero-one law

The Kolmogorowsche zero-one law , even zero-one law of Kolmogorov called and also in the alternative spellings Kolmogorov and Kolmogorov represented in the literature, is a mathematical theorem of probability theory about the possible chances of limits. It belongs to the zero-one laws and thus describes a class of events that are either almost certain (i.e., will occur with probability one) or are almost impossible (i.e. will occur with probability 0).

The law is named after Andrei Nikolayevich Kolmogorov .

formulation

A probability space and a sequence of σ-algebras in , i.e. for all, are given . If the σ-algebras are all stochastically independent of one another, then: ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle ({\mathcal {A}}_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}_{n}\subseteq {\mathcal {A}}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle {\mathcal {A}}_{n}}$

The terminal σ-algebra of the sequence is P-trivial , i.e. for every terminal event there is either or . ${\displaystyle {\mathcal {T}}}$ ${\displaystyle ({\mathcal {A}}_{n})_{n\in \mathbb {N} }}$ ${\displaystyle T\in {\mathcal {T}}}$ ${\displaystyle P(T)=0}$ ${\displaystyle P(T)=1}$

The same statement applies to the terminal σ-algebra of a sequence of stochastically independent random variables as well as to the terminal σ-algebra of a sequence of stochastically independent events .

Implications

Be independent random variables and to with terminal associated algebra. It is easy to show that it is true. So the sequence will almost certainly converge or diverge. In the first case , if it denotes the limit, it can also be shown that it is a measurable random variable. Since is trivial, it must necessarily be constant. ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle ({\mathcal {A}}_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {A}}_{n}=\sigma (X_{n})}$ ${\displaystyle \sigma }$ ${\displaystyle \{\omega \mid X_{n}(\omega )\;{\mbox{konvergiert für}}\;n\to \infty \}\in {\mathcal {T}}}$ ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle \sigma ({\mathcal {T}})}$ ${\displaystyle \sigma ({\mathcal {T}})}$ ${\displaystyle X}$

In addition, the zero-one law of Hewitt-Savage can be derived using Kolmogorow's zero-one law .

Evidence sketch

One defines

${\displaystyle {\mathcal {K}}_{n}:=\sigma \left(\bigcup _{i=1}^{n}{\mathcal {A}}_{i}\right){\text{ und }}{\mathcal {L}}_{n}:=\sigma \left(\bigcup _{i=n+1}^{\infty }{\mathcal {A}}_{i}\right)}$,

so:

${\displaystyle {\mathcal {K}}_{n}}$is independent of . ${\displaystyle {\mathcal {L}}_{n}}$

Furthermore, in included, so true ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {L}}_{n}}$

${\displaystyle {\mathcal {K}}_{n}}$is independent of for everyone . ${\displaystyle {\mathcal {T}}}$ ${\displaystyle n}$

Then it follows regardless of and because of the cutting stability ${\displaystyle \bigcup _{i=1}^{\infty }{\mathcal {K}}_{i}}$ ${\displaystyle {\mathcal {T}}}$

${\displaystyle {\mathcal {K}}_{\infty }:=\sigma \left(\bigcup _{i=1}^{\infty }{\mathcal {K}}_{i}\right)}$ is independent of ${\displaystyle {\mathcal {T}}}$

However, since in is included, followed ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {K}}_{\infty }}$

${\displaystyle {\mathcal {T}}}$is independent of , ${\displaystyle {\mathcal {T}}}$

from which it follows directly that P is trivial. ${\displaystyle {\mathcal {T}}}$

The proof for sequences of events or random variables follows analogously, since the terminal σ-algebra of events and random variables is defined as the terminal σ-algebra of the generated σ-algebras.

Generalizations

Kolmogorov's zero-one law is formulated more generally in the literature in the following ways:

• It is not formulated for sequences of independent σ-algebras and their terminal σ-algebra, but more generally for any set system . For the validity of the statement, however, in addition to the independence, the cutting stability of the set systems must also be required. Otherwise the statement remains unchanged.
• A conditional version is formulated with recourse to the conditional independence and the conditional probability, as defined by the conditional expected value . This means you bet

${\displaystyle P(A|{\mathcal {G}}):=\operatorname {E} (\mathbf {1} _{A}|{\mathcal {G}})}$

Then Kolmogorov's zero-one law reads:

If there is a sequence of conditionally independent, intersection-stable set systems and if the associated terminal σ-algebra is given, then: ${\displaystyle {\mathcal {T}}}$

• It is for everyone ${\displaystyle P(T_{1}|{\mathcal {G}})P(T_{2}|{\mathcal {G}})=P(T_{1}\cap T_{2}|{\mathcal {G}})}$ ${\displaystyle T_{1},T_{2}\in {\mathcal {T}}}$
• For each terminal numerical random variable there is a measurable random variable , so that applies. ${\displaystyle X}$ ${\displaystyle {\mathcal {G}}}$ ${\displaystyle Y}$ ${\displaystyle X=Y}$
• For every terminal event applies and there exists such that is. ${\displaystyle T}$ ${\displaystyle P(T|{\mathcal {G}})=\mathbf {1} _{T}}$ ${\displaystyle G\in {\mathcal {G}}}$ ${\displaystyle \mathbf {1} _{T}=\mathbf {1} _{G}}$

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .
• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , doi : 10.1007 / 978-3-642-21026-6 .

Individual evidence

1. ^ Schmidt: Measure and probability. 2011, p. 235.
2. ^ Schmidt: Measure and probability. 2011, p. 441.