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 $[0,\mathrm{\infty })$ starting from deterministic point has also deterministic initial movement.

Statement

Suppose that $X=(X_t:t\ge 0)$ is an adapted right continuous Feller process on a probability space $(\mathrm{\Omega },{ℱ},\{{{ℱ}}_t{\}}_{t\ge 0},\mathbb{\mathbb{P}})$ such that $X_0$ is constant with probability one. Let ${{ℱ}}_t^X:=\sigma (X_s;s\le t),{{ℱ}}_{t^{+}}^X:=\bigcap _{s>t}{{ℱ}}_s^X$. Then any event in the germ sigma algebra $\mathrm{\Lambda }\in {{ℱ}}_{0+}^X$ has either $\mathbb{\mathbb{P}}(\mathrm{\Lambda })=0$ or $\mathbb{\mathbb{P}}(\mathrm{\Lambda })=1.$

Generalization

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