Hewitt-Savage Zero One Law

The 0-1 law of Hewitt-Savage is a set of probability theory , like all zero-one law makes statements about when an event almost certain (ie with probability 1) occurs or is almost impossible (ie probability 0 owns) .

statement

Given a sequence of independently and identically distributed random variables and the interchangeable σ-algebra of the sequence. Then P is trivial , so it is either or for every event . ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle E\in {\mathcal {E}}}$ ${\displaystyle P(E)=0}$ ${\displaystyle P(E)=1}$

Derivation

The derivation is based on Kolmogorov's zero-one law . This means that the terminal σ-algebra of a sequence of independently identically distributed random variables is always P-trivial. Since, however, independently identically distributed random variables are always also exchangeable families of random variables , it then also applies to every exchangeable event that a terminal event exists, so that there is. The statement follows from this. ${\displaystyle E\in {\mathcal {E}}}$ ${\displaystyle B\in {\mathcal {T}}}$ ${\displaystyle P(E\,\triangle \,B)=0}$

literature

• Achim Klenke : Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 242 , doi : 10.1007 / 978-3-642-36018-3 .