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Approximation of Distributions by Bounded Sets

Approximation of Distributions by Bounded Sets Let P be a probability distribution on a locally compact separable metric space (S,d). We study the following problem of approximation of a distribution P by a set A from a given class $\mathcal{A}\subset2^{S}$ : $$W(A,P)\equiv\int_{S}\varphi(d(x,A))P(dx)\to\min_{A\in\mathcal{A}},$$ where φ is a nondecreasing function. A special case where $\mathcal{A}$ consists of unions of bounded sets, $\mathcal{A}=\{\bigcup_{i=1}^{k}A_{i}:\Delta(A_{i})\leq K,\ i=1,\ldots,k\}$ , is considered in detail. We give sufficient conditions for the existence of an optimal approximative set and for the convergence of the sequence of optimal sets A n found for measures P n which satisfy P n ⇒ P. Current article is a follow-up to Käärik and Pärna (Acta Appl. Math. 78, 175–183, 2003; Acta Comment. Univ. Tartu. 8, 101–112, 2004) where the case of parametric sets was studied. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Approximation of Distributions by Bounded Sets

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References (26)

Publisher
Springer Journals
Copyright
Copyright © 2007 by Springer Science + Business Media B.V.
Subject
Mathematics; Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Classical Mechanics
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-007-9140-z
Publisher site
See Article on Publisher Site

Abstract

Let P be a probability distribution on a locally compact separable metric space (S,d). We study the following problem of approximation of a distribution P by a set A from a given class $\mathcal{A}\subset2^{S}$ : $$W(A,P)\equiv\int_{S}\varphi(d(x,A))P(dx)\to\min_{A\in\mathcal{A}},$$ where φ is a nondecreasing function. A special case where $\mathcal{A}$ consists of unions of bounded sets, $\mathcal{A}=\{\bigcup_{i=1}^{k}A_{i}:\Delta(A_{i})\leq K,\ i=1,\ldots,k\}$ , is considered in detail. We give sufficient conditions for the existence of an optimal approximative set and for the convergence of the sequence of optimal sets A n found for measures P n which satisfy P n ⇒ P. Current article is a follow-up to Käärik and Pärna (Acta Appl. Math. 78, 175–183, 2003; Acta Comment. Univ. Tartu. 8, 101–112, 2004) where the case of parametric sets was studied.

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: Mar 27, 2007

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