A zero-one law for Gaussian processes
HTML articles powered by AMS MathViewer
- by Naresh C. Jain PDF
- Proc. Amer. Math. Soc. 29 (1971), 585-587 Request permission
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.References
- B. Jamison and S. Orey, Subgroups of sequences and paths, Proc. Amer. Math. Soc. 24 (1970), 739–744. MR 254914, DOI 10.1090/S0002-9939-1970-0254914-7
- G. Kallianpur, Zero-one laws for Gaussian processes, Trans. Amer. Math. Soc. 149 (1970), 199–211. MR 266293, DOI 10.1090/S0002-9947-1970-0266293-4
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 29 (1971), 585-587
- MSC: Primary 60.40
- DOI: https://doi.org/10.1090/S0002-9939-1971-0278369-2
- MathSciNet review: 0278369