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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


A zero-one law for Gaussian processes

Author: Naresh C. Jain
Journal: Proc. Amer. Math. Soc. 29 (1971), 585-587
MSC: Primary 60.40
MathSciNet review: 0278369
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Abstract: Let $ {P_0}$ be a Gaussian probability measure on the measurable space $ (X,B(X))$, where X is a linear space of realvalued functions over a complete separable metric space T, and $ B(X)$ is the $ \sigma $-algebra generated by sets of the form $ \{ x \in X:(x({t_1}), \cdots ,x({t_n})) \in {B^n}\} ;{B^n}$ being the Borel sets of $ {R^n},n \geqq 1$. The covariance $ R(s,t)$ is assumed continuous on $ T \times T$. If G is a subgroup of X and belongs to the $ \sigma $-algebra $ {B_0}(X)$ (the completion of $ B(X)$ with respect to $ {P_0}$), then it is shown that $ {P_0}(G) = 0$ or 1.

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PII: S 0002-9939(1971)0278369-2
Keywords: Gaussian process, zero-one law
Article copyright: © Copyright 1971 American Mathematical Society