publication . Other literature type . Article . Preprint . 2010

Continuous Disintegrations of Gaussian Processes

T LaGatta;
Open Access
  • Published: 04 Mar 2010
  • Publisher: Society for Industrial & Applied Mathematics (SIAM)
Abstract
The goal of this paper is to understand the conditional law of a stochastic process once it has been observed over an interval. To make this precise, we introduce the notion of a continuous disintegration: a regular conditional probability measure which varies continuously in the conditioned parameter. The conditioning is infinite dimensional in character, which leads us to consider the general case of probability measures in Banach spaces. Our main result is that for a certain quantity $M$ based on the covariance structure, $M < \oo$ is a necessary and sufficient condition for a Gaussian measure to have a continuous disintegration. The condition $M < \oo$ is qu...
Subjects
free text keywords: Statistics, Probability and Uncertainty, Statistics and Probability, Mathematics - Probability, Mathematics - Functional Analysis, 41A65, 47A50, Gaussian measure, Covariance, Probability measure, Conditioning, Applied mathematics, Stochastic process, Gaussian process, symbols.namesake, symbols, Mathematics, Regular conditional probability, Banach space, Statistical physics
16 references, page 1 of 2

[1] D.R. Bell. The Malliavin Calculus. Longman Scientific and Technical, 1987.

[2] L. Bergamaschi. Geostatistics in hydrology: Kriging interpolation. http://www.dmsa.unipd.it/~berga/Teaching/STAM/stat.pdf.

[3] A. Berlinet and C. Thomas-Agnan. Reproducing kernel Hilbert spaces in probability and statistics. Kluwer Academic Publishers, 2004. [OpenAIRE]

[4] P. Billingsley. Convergence of probability measures, volume 2333096. Wiley New York, 1968.

[5] V.I. Bogachev. Measure Theory Vol. I-II, 2007.

[6] J.T. Chang and D. Pollard. Conditioning as disintegration. Statistica Neerlandica, 51(3):287-317, 1997.

[7] R. Durrett. Probability: theory and examples. Duxbury Press Belmont, CA, 1996.

[8] GB Folland. Real Analysis: Modern Techniques and Their Applications. Wiley-Interscience, 1999.

[9] L. Gross. Abstract Wiener spaces. In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probablility, volume 2, 1964.

[10] S. Janson. Gaussian Hilbert Spaces. Cambridge University Press, 1997.

[11] D. Le˜ao Jr, M. Fragoso, and P. Ruffino. Regular conditional probability, disintegration of probability and Radon spaces. Proyecciones (Antofagasta), 23(1), 2004. [OpenAIRE]

[12] M. Reed and B. Simon. Methods of Modern Mathematical Physics. Academic Press, 1981.

[13] V. Tarieladze and N. Vakhania. Disintegration of Gaussian measures and average-case optimal algorithms. Journal of Complexity, 23(4-6):851-866, 2007. [OpenAIRE]

[14] Mark Meckes (MO user 1044). Convergence of gaussian measures. MathOverflow. URL: http://mathoverflow.net/questions/16518 (v2010-02-26).

[15] NN Vakhania. The topological support of Gaussian measure in Banach space. Nagoya Math. J, 57:59-63, 1975. [OpenAIRE]

16 references, page 1 of 2
Any information missing or wrong?Report an Issue