论文信息 - GAUSSIAN MEASURES IN FUNCTION SPACE

GAUSSIAN MEASURES IN FUNCTION SPACE

Two Gaussian measures are either mutually singular or equivalent. This dichotomy was first discovered by Feldman and Hajek (independently). We give a simple, almost formal, proof of this result, based on the study of a certain pair of functionals of the two measures. In addition we show that two Gaussian measures with zero means and smooth Polya-type covariances (on an interval) are equivalent if and only if the right-hand slopes of the covariances at zero are equal. The H and J functionals* Two probability measures μ0 and μx on a space (Ω, &) are called mutually singular (μ0 _L μd if there is a set ΰe & for which μo(B) — 0 and μλ{Ω — B) = 0. The measures are called mutually equivalent (μ0 ~ μ ±) if they have the same zero sets, i.e., μo(B) = 0 if and only if μ^B) = 0. Setting μ = μ0 + μx we may define the Radon-Nikodym derivatives,

L. Shepp | Lawrence A. Shepp

[1] C. Radhakrishna Rao,et al. DISCRIMINATION OF GAUSSIAN PROCESSES , 1965 .

[2] S. Kakutani. On Equivalence of Infinite Product Measures , 1948 .

[3] Jaroslav Hájek. A property of $J$-divergences of marginal probability distributions , 1958 .

[4] C. Kraft. Some conditions for consistency and uniform consistency of statistical procedures , 1955 .

[5] Ernst Hellinger,et al. Die Orthogonalinvarianten quadratischer Formen von unendlichvielen Variabelen , 2022 .

[6] Jacob Feldman,et al. Equivalence and perpendicularity of Gaussian processes , 1958 .

[7] U. Grenander. Stochastic processes and statistical inference , 1950 .

[8] H. Jeffreys,et al. Theory of probability , 1896 .

[9] L. Shepp,et al. THE SINGULARITY OF GAUSSIAN MEASURES IN FUNCTION SPACE. , 1964, Proceedings of the National Academy of Sciences of the United States of America.

[10] Lester L. Helms,et al. Mean convergence of martingales , 1958 .

[11] J. Doob. Stochastic processes , 1953 .

[12] G. Baxter,et al. A strong limit theorem for Gaussian processes , 1956 .

[13] H. D. Parker,et al. The Bell Telephone Laboratories, Inc. , 1925, Journal of the A.I.E.E..