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 $(\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t\geq 0},\mathbb {P} )$ such that $X_{0}$ is constant with probability one. Let ${\mathcal {F}}_{t}^{X}:=\sigma (X_{s};s\leq t),{\mathcal {F}}_{t^{+}}^{X}:=\bigcap _{s>t}{\mathcal {F}}_{s}^{X}$. Then any event in the germ sigma algebra $\Lambda \in {\mathcal {F}}_{0+}^{X}$ has either $\mathbb {P} (\Lambda )=0$ or $\mathbb {P} (\Lambda )=1.$

Generalization

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