In the mathematical theory of probability, Blumenthal's zero–one law,[1] 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
starting from deterministic point has also deterministic initial movement.
Statement
Suppose that
is an adapted right continuous Feller process on a probability space
such that
is constant with probability one. Let
. Then any event in the germ sigma algebra
has either
or
Generalization
Suppose that
is an adapted stochastic process on a probability space
such that
is constant with probability one. If
has Markov property with respect to the filtration
then any event
has either
or
Note that every right continuous Feller process on a probability space
has strong Markov property with respect to the filtration
.
References
- ^ Blumenthal, Robert M. (1957), "An extended Markov property", Transactions of the American Mathematical Society, 85 (1): 52–72, doi:10.1090/s0002-9947-1957-0088102-2, JSTOR 1992961, MR 0088102, Zbl 0084.13602