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 [math]\displaystyle{ [0,\infty) }[/math] starting from deterministic point has also deterministic initial movement.

Statement

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

Generalization

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