Blumenthal's zero–one law

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 $[0,\infty )$ with the strong Markov property has an essentially deterministic starting point.

Statement

Suppose that $X=(X_{t}:t\geq 0)$ is a stochastic process on a probability space $(\Omega ,{\mathcal {F}},\mathbb {P} )$ with $\Omega =C(\mathbb {R} ^{+},\mathbb {R} )$ , natural filtration $\{{\mathcal {F}}_{t}\}_{{t\geq 0}}$ and canonical identification $X_{t}(\omega )=\omega (t)$ . If $X$ has the strong Markov property then any event in the germ sigma algebra $\Lambda \in {\mathcal {F}}_{0+}=\bigcap _{s>0}{\mathcal {F}}_{s}$ has either $\mathbb {P} (\Lambda )=0$ or $\mathbb {P} (\Lambda )=1.$ ^[2]

References

↑ Blumenthal, Robert M. (1957), "An extended Markov property", Transactions of the American Mathematical Society, 85: 52–72, doi:10.2307/1992961, JSTOR 1992961, MR 0088102, Zbl 0084.13602
↑ Rogers, L. C. G.; Williams, D. (2000), Diffusions, Markov Processes, and Martingales, Cambridge University Press, pp. 23, §I.12, ISBN 0521775949, OCLC 42874839

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[1] Blumenthal, Robert M. (1957), "An extended Markov property", Transactions of the American Mathematical Society, 85: 52–72, doi:10.2307/1992961, 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 0521775949, OCLC 42874839