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Blumenthal's zero–one law

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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 memoryless processes. Loosely, it states that any stochastic process on with the strong Markov property has an essentially deterministic starting point.

Statement

Suppose that is a stochastic process on a probability space with , natural filtration and canonical identification . If has the strong Markov property then any event in the germ sigma algebra has either or [2]

Heuristic explanation

The above statement means roughly speaking that the initial datum of the process is deterministic. Indeed, the law of the random variable satisfies for all -measurable event . Such a probability measure is necessarily of the form for some particular point , and as a consequence for -almost all paths . In other words, the process starts almost surely from the completely deterministic initial position .

References

  1. ^ 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
  2. ^ Rogers, L. C. G.; Williams, D. (2000), Diffusions, Markov Processes, and Martingales, Cambridge University Press, pp. 23, §I.12, ISBN 978-0521775946, OCLC 42874839