Blumenthal's zero–one law explained

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 , inf ty) 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, l{F},\{ l{F} _{t} \} _{t\geq} , P ) such that X _{0} is constant with probability one. Let l{F} ^{X} _{t: = \sigma(X} _{s;} s\leq t), l{F} ^{X} _{t ^{+}} : = c ap _{s>t} l{F} ^{X} _{s} . Then any event in the germ sigma algebra Λ \in l{F} ^{X} _{0+} has either P ( Λ ) =0 or P ( Λ ) =1 .

Generalization

Suppose that

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