Blumenthal's Zero–one Law

In the mathematical theory of probability, Blumenthal's zero–one law, named after Robert McCallum Blumenthal, is a statement about the nature of the beginnings of right continuous Feller process. Loosely, it states that any right continuous Feller process on $[0,\infty)$ starting from deterministic point has also deterministic initial movement.

Statement

Suppose that $X=(X_t:t\geq 0)$ is an adapted right continuous Feller process on a probability space $(\Omega,\mathcal,\_,\mathbb)$ such that $X_0$ is constant with probability one. Let $\mathcal^X_t:=\sigma(X_s; s\leq t), \mathcal^X_:=\bigcap_\mathcal^X_s$. Then any event in the germ sigma algebra $\Lambda \in \mathcal^X_$ has either $\mathbb(\Lambda)=0$ or $\mathbb(\Lambda)=1.$

Generalization

Suppose that $X=(X_t:t\geq 0)$ is an adapted stochastic process on a probability space $(\Omega,\mathcal,\_,\mathbb)$ such that $X_0$ is constant with probability one. If $X$ has Markov property with respect to the filtration $\_$ then any event $\Lambda \in \mathcal^X_$ has either $\mathbb(\Lambda)=0$ or $\mathbb(\Lambda)=1.$ Note that every right continuous Feller process on a probability space $(\Omega,\mathcal,\_,\mathbb)$ has strong Markov property with respect to the filtration $\_$.

References

{{DEFAULTSORT:Blumenthal's zero-one law
Probability theory