Blumenthal's zero-one law

The Blumenthal 0-1 Law , named after RM Blumenthal, is a mathematical theorem in the field of probability theory . Like all zero-one laws , it describes a class of events whose probabilities are always 0 or 1.

statement

Let be a probability space and a defined Brownian motion with filtration . Then the σ-algebra defined by , is -trivial , i.e. H. it applies: for everyone . ${\displaystyle (\Omega ,{\mathcal {A}},\mathbb {P} )}$ ${\displaystyle (B_{t})_{t\geq 0}}$ ${\displaystyle {\mathcal {F}}_{t}=\sigma (\{B_{s}\mid s\leq t\})}$ ${\displaystyle {\mathcal {F}}_{0}^{+}}$ ${\displaystyle \textstyle {\mathcal {F}}_{0}^{+}=\bigcap _{t>0}{\mathcal {F}}_{t}}$ ${\displaystyle \mathbb {P} }$ ${\displaystyle \mathbb {P} (A)\in \{0,1\}}$ ${\displaystyle A\in {\mathcal {F}}_{0}^{+}}$

Descriptive includes exactly those events that only depend on , for anything small . For example, the event is , so it applies . ${\displaystyle {\mathcal {F}}_{0}^{+}}$ ${\displaystyle (B_{t})_{0\leq t\leq \varepsilon }}$ ${\displaystyle \varepsilon }$ ${\displaystyle A=\{\forall \varepsilon >0\exists t>0:\;t<\varepsilon \wedge B_{t}=0\}\in {\mathcal {F}}_{0}^{+}}$ ${\displaystyle \mathbb {P} (A)\in \{0,1\}}$

literature

• Blumenthal, RM: An extended Markov property. In: Transactions of the American Mathematical Society. Volume 85, 1957, pp. 52-72.
• Klenke, Achim: Probability Theory , Springer-Verlag Berlin Heidelberg 2008, ISBN 978-3-540-76317-8