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A moment problem for random discrete measures / Yuri G. Kondratiev, Tobias Kuna, Eugene Lytvynov

Stochastic Processes and their Applications, Volume: 125, Issue: 9, Pages: 3541 - 3569

Swansea University Author: Eugene Lytvynov

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DOI (Published version): 10.1016/j.spa.2015.03.007

Abstract

Let $X$ be a locally compact Polish space. A random measure on $X$ is a probability measure on the space of all (nonnegative) Radon measures on $X$.Denote by $\mathbb K(X)$ the cone of all Radon measures $\eta$ on $X$ which are of the form $\eta=\sum_{i}s_i\delta_{x_i}$, where, for each $i$, $s_i>...

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Published in: Stochastic Processes and their Applications
Published: 2015
URI: https://cronfa.swan.ac.uk/Record/cronfa22142
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Abstract: Let $X$ be a locally compact Polish space. A random measure on $X$ is a probability measure on the space of all (nonnegative) Radon measures on $X$.Denote by $\mathbb K(X)$ the cone of all Radon measures $\eta$ on $X$ which are of the form $\eta=\sum_{i}s_i\delta_{x_i}$, where, for each $i$, $s_i>0$ and $\delta_{x_i}$ is the Dirac measure at $x_i\in X$. A random discrete measure on $X$ is a probability measure on $\mathbb K(X)$. The main result of the paper states a necessary and sufficient condition (conditional upon a mild {\it a priori\/} bound) when a random measure $\mu$ is also a random discrete measure. This condition is formulated solely in terms of moments of the random measure $\mu$. Classical examples of random discrete measures are completely random measures and additive subordinators, however, the main result holds independently of any independence property. As a corollary, a characterisation via a moments is given when a random measure is a point process.
Keywords: Discrete random measure, moment problem, point process, random measure.
College: College of Science
Issue: 9
Start Page: 3541
End Page: 3569