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Random Operators and Stochastic Equations

Editor-in-Chief: Girko, Vyacheslav

Managing Editor: Molchanov, S.

Editorial Board: Accardi, L. / Albeverio, Sergio / Carmona, R. / Casati, G. / Christopeit, N. / Domanski, C. / Drygas, Hilmar / Gupta, A.K. / Ibragimov, I. / Kirsch, Werner / Klein, A. / Kondratyev, Yuri / Kurotschka, V. / Leonenko, N. / Loubaton, Philippe / Orsingher, E. / Pastur, L. / Rodrigues, Waldyr A. / Shiryaev, Albert / Turbin, A.F. / Veretennikov, Alexandre

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1569-397X
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Volume 19, Issue 3

Issues

Use of the global implicit function theorem to induce singular conditional distributions on surfaces in n dimensions: Part III

Donald Bamber / I. R. Goodman
  • Barracks Building 344, Space and Naval Warfare Systems Center, Pacific, San Diego, CA 92152, USA.
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/ Arjun K. Gupta / Hung T. Nguyen
Published Online: 2011-08-22 | DOI: https://doi.org/10.1515/ROSE.2011.013

Abstract

This paper stems from previous work of certain of the authors, where the issue of inducing distributions on lower dimensional spaces arose as a natural outgrowth of the main goal: the estimation of conditional probabilities, given other partially specified conditional probabilities as a premise set in a probability logic framework. This paper is concerned with the following problem. Let 1 ≤ m < n be fixed positive integers, some open domain, and a function yielding a full partitioning of D into a family, denoted M(h), of lower-dimensional surfaces/manifolds via inverse mapping h –1 as D = ⋃M(h), where M(h) = d{h –1(t) : t in range(h)}, noting each h –1(t) can also be considered the solution set of all X in D of the simultaneous equations h(X) = t. Let X be a random vector (rv) over D having a probability density function (pdf) ƒ. Then, if we add sufficient smoothness conditions concerning the behavior of h (continuous differentiability, full rank Jacobian matrix dh(X)/dX over D, etc.), can an explicit elementary approach be found for inducing from the full absolutely continuous distribution of X over D a necessarily singular distribution for X restricted to be over M(h) that satisfies a list of natural desirable properties? More generally, for fixed positive integer r, we can pose a similar question concerning rv ψ(X), when is some bounded a.e. continuous function, not necessarily admitting a pdf.

Keywords.: Global implicit function theorem; surfaces; hypersurfaces; general surfaces; distributions on hypersurfaces; distributions on surfaces; singular distributions; conditional expectations; conditional probability measures; surface integrals; surface measures; surface densities; geometric measure; geometric probability

About the article

Received: 2009-06-01

Accepted: 2009-12-12

Published Online: 2011-08-22

Published in Print: 2011-09-01


Citation Information: Random Operators and Stochastic Equations, Volume 19, Issue 3, Pages 217–265, ISSN (Online) 1569-397x, ISSN (Print) 0926-6364, DOI: https://doi.org/10.1515/ROSE.2011.013.

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