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Variational Analysis of Poisson Processes

Chapter in book
Authors Ilya Molchanov
Sergei Zuyev
Published in Stochastic Analysis for Poisson Point Processes. Giovanni Peccati; Matthias Reitzner (eds.)
Pages 81-101
ISBN 978-3-319-05232-8
ISSN 2039-1471
Publisher Springer
Publication year 2016
Published at Department of Mathematical Sciences
Pages 81-101
Language en
Links doi.org/10.1007/978-3-319-05233-5_3
Subject categories Mathematics

Abstract

© 2016 Springer International Publishing Switzerland.The expected value of a functional F(η) of a Poisson process η can be considered as a function of its intensity measure μ. The paper surveys several results concerning differentiability properties of this functional on the space of signed measures with finite total variation. Then, necessary conditions for μ being a local minima of the considered functional are elaborated taking into account possible constraints on μ, most importantly the case of μ with given total mass a. These necessary conditions can be phrased by requiring that the gradient of the functional (being the expected first difference) is constant on the support of μ. In many important cases, the gradient depends only on the local structure of μ in a neighbourhood of x and so it is possible to work out the asymptotics of the minimising measure with the total mass a growing to infinity. Examples include the optimal approximation of convex functions, clustering problem and optimal search. In non-asymptotic cases, it is in general possible to find the optimal measure using steepest descent algorithms which are based on the obtained explicit form of the gradient.

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Utskriftsdatum: 2019-11-20