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A moment problem for random discrete measures

Yuri G. Kondratiev, Tobias Kuna, Eugene Lytvynov Orcid Logo

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

Swansea University Author: Eugene Lytvynov Orcid Logo

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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
first_indexed 2015-06-23T02:07:51Z
last_indexed 2019-08-09T15:05:30Z
id cronfa22142
recordtype SURis
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spelling 2019-08-06T08:41:50.8446144 v2 22142 2015-06-22 A moment problem for random discrete measures e5b4fef159d90a480b1961cef89a17b7 0000-0001-9685-7727 Eugene Lytvynov Eugene Lytvynov true false 2015-06-22 MACS 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. Journal Article Stochastic Processes and their Applications 125 9 3541 3569 Discrete random measure, moment problem, point process, random measure. 31 12 2015 2015-12-31 10.1016/j.spa.2015.03.007 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 2019-08-06T08:41:50.8446144 2015-06-22T16:05:39.6348203 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Yuri G. Kondratiev 1 Tobias Kuna 2 Eugene Lytvynov 0000-0001-9685-7727 3
title A moment problem for random discrete measures
spellingShingle A moment problem for random discrete measures
Eugene Lytvynov
title_short A moment problem for random discrete measures
title_full A moment problem for random discrete measures
title_fullStr A moment problem for random discrete measures
title_full_unstemmed A moment problem for random discrete measures
title_sort A moment problem for random discrete measures
author_id_str_mv e5b4fef159d90a480b1961cef89a17b7
author_id_fullname_str_mv e5b4fef159d90a480b1961cef89a17b7_***_Eugene Lytvynov
author Eugene Lytvynov
author2 Yuri G. Kondratiev
Tobias Kuna
Eugene Lytvynov
format Journal article
container_title Stochastic Processes and their Applications
container_volume 125
container_issue 9
container_start_page 3541
publishDate 2015
institution Swansea University
doi_str_mv 10.1016/j.spa.2015.03.007
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 0
active_str 0
description 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.
published_date 2015-12-31T18:46:50Z
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score 11.047674

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