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Approximation of a free Poisson process by systems of freely independent particles
Marek Bożejko,
José Luís da Silva,
Tobias Kuna,
Eugene Lytvynov 
Infinite Dimensional Analysis, Quantum Probability and Related Topics, Volume: 21, Issue: 3, Pages: 1850020-1 - 1850020-25
Swansea University Author:
Eugene Lytvynov
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DOI (Published version): 10.1142/S0219025718500200
Abstract
Let $\sigma$ be a non-atomic, infinite Radon measure on $\mathbb R^d$, for example, $d\sigma(x)=z\,dx$ where $z>0$. We consider a system of freely independent particles $x_1,\dots,x_N$ in a bounded set $\Lambda\subset\mathbb R^d$, where each particle $x_i$ has distribution $\frac1{\sigma(\Lambda)...
| Published in: | Infinite Dimensional Analysis, Quantum Probability and Related Topics |
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| ISSN: | 0219-0257 1793-6306 |
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2018
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa43227 |
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<?xml version="1.0"?><rfc1807><datestamp>2018-10-08T14:54:03.1570657</datestamp><bib-version>v2</bib-version><id>43227</id><entry>2018-08-07</entry><title>Approximation of a free Poisson process by systems of freely independent particles</title><swanseaauthors><author><sid>e5b4fef159d90a480b1961cef89a17b7</sid><ORCID>0000-0001-9685-7727</ORCID><firstname>Eugene</firstname><surname>Lytvynov</surname><name>Eugene Lytvynov</name><active>true</active><ethesisStudent>false</ethesisStudent></author></swanseaauthors><date>2018-08-07</date><deptcode>SMA</deptcode><abstract>Let $\sigma$ be a non-atomic, infinite Radon measure on $\mathbb R^d$, for example, $d\sigma(x)=z\,dx$ where $z>0$. We consider a system of freely independent particles $x_1,\dots,x_N$ in a bounded set $\Lambda\subset\mathbb R^d$, where each particle $x_i$ has distribution $\frac1{\sigma(\Lambda)}\,\sigma$ on $\Lambda$ and the number of particles, $N$, is random and has Poisson distribution with parameter $\sigma(\Lambda)$. If the particles were classically independent rather than freely independent, this particle system would be the restriction to $\Lambda$ of the Poisson point process on $\mathbb R^d$ with intensity measure $\sigma$. In the case of free independence, this particle system is not the restriction of the free Poisson process on $\mathbb R^d$ with intensity measure $\sigma$. Nevertheless, we prove that this is true in an approximative sense: if bounded sets $\Lambda^{(n)}$ ($n\in\mathbb N$) are such that $\Lambda^{(1)}\subset\Lambda^{(2)}\subset\Lambda^{(3)}\subset\dotsm$ and $\bigcup_{n=1}^\infty \Lambda^{(n)}=\mathbb R^d$, then the corresponding particle system in $\Lambda^{(n)}$ converges (as $n\to\infty$) to the free Poisson process on $\mathbb R^d$ with intensity measure $\sigma$. We also prove the following $N/V$-limit: Let $N^{(n)}$ be a determinstic sequence of natural numbers such that $\lim_{n\to\infty}N^{(n)}/\sigma(\Lambda^{(n)})=1$. Then the system of $N^{(n)}$ freely independent particles in $\Lambda^{(n)}$ converges (as $n\to\infty$) to the free Poisson process. We finally extend these results to the case of a free L\'evy white noise (in particular, a free L\'evy process) without free Gaussian part.</abstract><type>Journal Article</type><journal>Infinite Dimensional Analysis, Quantum Probability and Related Topics</journal><volume>21</volume><journalNumber>3</journalNumber><paginationStart>1850020-1</paginationStart><paginationEnd>1850020-25</paginationEnd><publisher/><issnPrint>0219-0257</issnPrint><issnElectronic>1793-6306</issnElectronic><keywords/><publishedDay>12</publishedDay><publishedMonth>9</publishedMonth><publishedYear>2018</publishedYear><publishedDate>2018-09-12</publishedDate><doi>10.1142/S0219025718500200</doi><url/><notes/><college>COLLEGE NANME</college><department>Mathematics</department><CollegeCode>COLLEGE CODE</CollegeCode><DepartmentCode>SMA</DepartmentCode><institution>Swansea University</institution><apcterm/><lastEdited>2018-10-08T14:54:03.1570657</lastEdited><Created>2018-08-07T10:24:32.0454924</Created><path><level id="1">Faculty of Science and Engineering</level><level id="2">School of Mathematics and Computer Science - Mathematics</level></path><authors><author><firstname>Marek</firstname><surname>Bożejko</surname><order>1</order></author><author><firstname>José Luís</firstname><surname>da Silva</surname><order>2</order></author><author><firstname>Tobias</firstname><surname>Kuna</surname><order>3</order></author><author><firstname>Eugene</firstname><surname>Lytvynov</surname><orcid>0000-0001-9685-7727</orcid><order>4</order></author></authors><documents><document><filename>0043227-07082018102716.pdf</filename><originalFilename>Free_Poisson_approximation.pdf</originalFilename><uploaded>2018-08-07T10:27:16.8600000</uploaded><type>Output</type><contentLength>312172</contentLength><contentType>application/pdf</contentType><version>Accepted Manuscript</version><cronfaStatus>true</cronfaStatus><embargoDate>2019-09-12T00:00:00.0000000</embargoDate><copyrightCorrect>true</copyrightCorrect><language>eng</language></document></documents><OutputDurs/></rfc1807> |
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2018-10-08T14:54:03.1570657 v2 43227 2018-08-07 Approximation of a free Poisson process by systems of freely independent particles e5b4fef159d90a480b1961cef89a17b7 0000-0001-9685-7727 Eugene Lytvynov Eugene Lytvynov true false 2018-08-07 SMA Let $\sigma$ be a non-atomic, infinite Radon measure on $\mathbb R^d$, for example, $d\sigma(x)=z\,dx$ where $z>0$. We consider a system of freely independent particles $x_1,\dots,x_N$ in a bounded set $\Lambda\subset\mathbb R^d$, where each particle $x_i$ has distribution $\frac1{\sigma(\Lambda)}\,\sigma$ on $\Lambda$ and the number of particles, $N$, is random and has Poisson distribution with parameter $\sigma(\Lambda)$. If the particles were classically independent rather than freely independent, this particle system would be the restriction to $\Lambda$ of the Poisson point process on $\mathbb R^d$ with intensity measure $\sigma$. In the case of free independence, this particle system is not the restriction of the free Poisson process on $\mathbb R^d$ with intensity measure $\sigma$. Nevertheless, we prove that this is true in an approximative sense: if bounded sets $\Lambda^{(n)}$ ($n\in\mathbb N$) are such that $\Lambda^{(1)}\subset\Lambda^{(2)}\subset\Lambda^{(3)}\subset\dotsm$ and $\bigcup_{n=1}^\infty \Lambda^{(n)}=\mathbb R^d$, then the corresponding particle system in $\Lambda^{(n)}$ converges (as $n\to\infty$) to the free Poisson process on $\mathbb R^d$ with intensity measure $\sigma$. We also prove the following $N/V$-limit: Let $N^{(n)}$ be a determinstic sequence of natural numbers such that $\lim_{n\to\infty}N^{(n)}/\sigma(\Lambda^{(n)})=1$. Then the system of $N^{(n)}$ freely independent particles in $\Lambda^{(n)}$ converges (as $n\to\infty$) to the free Poisson process. We finally extend these results to the case of a free L\'evy white noise (in particular, a free L\'evy process) without free Gaussian part. Journal Article Infinite Dimensional Analysis, Quantum Probability and Related Topics 21 3 1850020-1 1850020-25 0219-0257 1793-6306 12 9 2018 2018-09-12 10.1142/S0219025718500200 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2018-10-08T14:54:03.1570657 2018-08-07T10:24:32.0454924 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Marek Bożejko 1 José Luís da Silva 2 Tobias Kuna 3 Eugene Lytvynov 0000-0001-9685-7727 4 0043227-07082018102716.pdf Free_Poisson_approximation.pdf 2018-08-07T10:27:16.8600000 Output 312172 application/pdf Accepted Manuscript true 2019-09-12T00:00:00.0000000 true eng |
| title |
Approximation of a free Poisson process by systems of freely independent particles |
| spellingShingle |
Approximation of a free Poisson process by systems of freely independent particles Eugene Lytvynov |
| title_short |
Approximation of a free Poisson process by systems of freely independent particles |
| title_full |
Approximation of a free Poisson process by systems of freely independent particles |
| title_fullStr |
Approximation of a free Poisson process by systems of freely independent particles |
| title_full_unstemmed |
Approximation of a free Poisson process by systems of freely independent particles |
| title_sort |
Approximation of a free Poisson process by systems of freely independent particles |
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e5b4fef159d90a480b1961cef89a17b7 |
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e5b4fef159d90a480b1961cef89a17b7_***_Eugene Lytvynov |
| author |
Eugene Lytvynov |
| author2 |
Marek Bożejko José Luís da Silva Tobias Kuna Eugene Lytvynov |
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Journal article |
| container_title |
Infinite Dimensional Analysis, Quantum Probability and Related Topics |
| container_volume |
21 |
| container_issue |
3 |
| container_start_page |
1850020-1 |
| publishDate |
2018 |
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Swansea University |
| issn |
0219-0257 1793-6306 |
| doi_str_mv |
10.1142/S0219025718500200 |
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Faculty of Science and Engineering |
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Faculty of Science and Engineering |
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School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics |
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| description |
Let $\sigma$ be a non-atomic, infinite Radon measure on $\mathbb R^d$, for example, $d\sigma(x)=z\,dx$ where $z>0$. We consider a system of freely independent particles $x_1,\dots,x_N$ in a bounded set $\Lambda\subset\mathbb R^d$, where each particle $x_i$ has distribution $\frac1{\sigma(\Lambda)}\,\sigma$ on $\Lambda$ and the number of particles, $N$, is random and has Poisson distribution with parameter $\sigma(\Lambda)$. If the particles were classically independent rather than freely independent, this particle system would be the restriction to $\Lambda$ of the Poisson point process on $\mathbb R^d$ with intensity measure $\sigma$. In the case of free independence, this particle system is not the restriction of the free Poisson process on $\mathbb R^d$ with intensity measure $\sigma$. Nevertheless, we prove that this is true in an approximative sense: if bounded sets $\Lambda^{(n)}$ ($n\in\mathbb N$) are such that $\Lambda^{(1)}\subset\Lambda^{(2)}\subset\Lambda^{(3)}\subset\dotsm$ and $\bigcup_{n=1}^\infty \Lambda^{(n)}=\mathbb R^d$, then the corresponding particle system in $\Lambda^{(n)}$ converges (as $n\to\infty$) to the free Poisson process on $\mathbb R^d$ with intensity measure $\sigma$. We also prove the following $N/V$-limit: Let $N^{(n)}$ be a determinstic sequence of natural numbers such that $\lim_{n\to\infty}N^{(n)}/\sigma(\Lambda^{(n)})=1$. Then the system of $N^{(n)}$ freely independent particles in $\Lambda^{(n)}$ converges (as $n\to\infty$) to the free Poisson process. We finally extend these results to the case of a free L\'evy white noise (in particular, a free L\'evy process) without free Gaussian part. |
| published_date |
2018-09-12T03:54:30Z |
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1763752720506814464 |
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10.999161 |