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An extension of Kolmogorov's theorem for continuous covariances

Author: G. D. Allen
Journal: Proc. Amer. Math. Soc. 39 (1973), 214-216
MSC: Primary 60G05; Secondary 46C10
MathSciNet review: 0312554
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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 [Enhancements On Off] (What's this?)

  • [1] 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.
  • [2] 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.
  • [3] Yu. A. Rozanov, Stationary random processes, Translated from the Russian by A. Feinstein, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1967. MR 0214134

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Keywords: Representation of stochastic processes, covariance function, representation of covariance function
Article copyright: © Copyright 1973 American Mathematical Society