An extension of Kolmogorov’s theorem for continuous covariances
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- by G. D. Allen PDF
- Proc. Amer. Math. Soc. 39 (1973), 214-216 Request permission
Abstract:
The theorem of Kolmogorov stating that a non-negative definite kernel on ${N^1} \times {N^{ - 1}}$ is the covariance of a stochastic process on ${N^1}$ is generalized to continuous nonnegative definite functions on $Y \times Y,Y$ being a separable Hausdorff space. Also, a representation of such continuous nonnegative definite functions and their associated stochastic processes is provided.References
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D. K. Faddeev and V. N. Faddeeva, Computational methods in linear algebra, Fizmatgiz, Moscow, 1960; English transi., Freeman, San Francisco, Calif., 1963. MR 28 #1742; #4659.
A. N. Kolmogorov, Stationary sequences in Hilbert space, Byull. Moskov. Gos. Univ. Mat. 2 (1941), no. 6, 1-40; English transl, by Natasha Artin. MR 5,101 ; MR 13, 1138.
- Yu. A. Rozanov, Stationary random processes, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1967. Translated from the Russian by A. Feinstein. MR 0214134
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 39 (1973), 214-216
- MSC: Primary 60G05; Secondary 46C10
- DOI: https://doi.org/10.1090/S0002-9939-1973-0312554-8
- MathSciNet review: 0312554