Convergence Towards the End Space for Random Walks on Schreier Graphs
We consider a transitive action of a finitely generated group G and the Schreier graph Γ defined by this action for some fixed generating set. For a probability measure μ on G with a finite first moment, we show that if the induced random walk is transient, it converges towards the space of ends of...
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| Veröffentlicht in: | Journal of theoretical probability 2022-09, Vol.35 (3), p.1412-1422 |
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| container_title | Journal of theoretical probability |
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| creator | Stankov, Bogdan |
| description | We consider a transitive action of a finitely generated group
G
and the Schreier graph
Γ
defined by this action for some fixed generating set. For a probability measure
μ
on
G
with a finite first moment, we show that if the induced random walk is transient, it converges towards the space of ends of
Γ
. As a corollary, we obtain that for a probability measure with a finite first moment on Thompson’s group
F
, the support of which generates
F
as a semigroup, the induced random walk on the dyadic numbers has a non-trivial Poisson boundary. Some assumption on the moment of the measure is necessary as follows from an example by Juschenko and Zheng. |
| doi_str_mv | 10.1007/s10959-021-01104-6 |
| format | Article |
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G
and the Schreier graph
Γ
defined by this action for some fixed generating set. For a probability measure
μ
on
G
with a finite first moment, we show that if the induced random walk is transient, it converges towards the space of ends of
Γ
. As a corollary, we obtain that for a probability measure with a finite first moment on Thompson’s group
F
, the support of which generates
F
as a semigroup, the induced random walk on the dyadic numbers has a non-trivial Poisson boundary. Some assumption on the moment of the measure is necessary as follows from an example by Juschenko and Zheng.</description><identifier>ISSN: 0894-9840</identifier><identifier>EISSN: 1572-9230</identifier><identifier>DOI: 10.1007/s10959-021-01104-6</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Convergence ; Mathematics ; Mathematics and Statistics ; Probability Theory and Stochastic Processes ; Random walk ; Statistics</subject><ispartof>Journal of theoretical probability, 2022-09, Vol.35 (3), p.1412-1422</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021</rights><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-73058f9edbdbee99992f533732b1006cb1492a0e394b21a30caed3e6a4540c5b3</cites><orcidid>0000-0003-1803-6629</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10959-021-01104-6$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10959-021-01104-6$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27923,27924,41487,42556,51318</link.rule.ids></links><search><creatorcontrib>Stankov, Bogdan</creatorcontrib><title>Convergence Towards the End Space for Random Walks on Schreier Graphs</title><title>Journal of theoretical probability</title><addtitle>J Theor Probab</addtitle><description>We consider a transitive action of a finitely generated group
G
and the Schreier graph
Γ
defined by this action for some fixed generating set. For a probability measure
μ
on
G
with a finite first moment, we show that if the induced random walk is transient, it converges towards the space of ends of
Γ
. As a corollary, we obtain that for a probability measure with a finite first moment on Thompson’s group
F
, the support of which generates
F
as a semigroup, the induced random walk on the dyadic numbers has a non-trivial Poisson boundary. Some assumption on the moment of the measure is necessary as follows from an example by Juschenko and Zheng.</description><subject>Convergence</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Probability Theory and Stochastic Processes</subject><subject>Random walk</subject><subject>Statistics</subject><issn>0894-9840</issn><issn>1572-9230</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp9kEFLAzEQhYMoWKt_wFPAc3SSbDabo5RahYJgKx5Ddne229pu1qRV_PdGV_DmXAaG997wPkIuOVxzAH0TORhlGAjOgHPIWH5ERlxpwYyQcExGUJiMmSKDU3IW4wYAjAEYkenEd-8YVthVSJf-w4U60n2LdNrVdNG7dG18oE-uq_2Ovrjta6S-o4uqDbjGQGfB9W08JyeN20a8-N1j8nw3XU7u2fxx9jC5nbNKaNgzLUEVjcG6rEtEk0Y0SkotRZla5FXJMyMcoDRZKbiTUDmsJeYuUxlUqpRjcjXk9sG_HTDu7cYfQpdeWqF5IYTSWiWVGFRV8DEGbGwf1jsXPi0H-43LDrhswmV_cNk8meRgikncrTD8Rf_j-gI7zWvA</recordid><startdate>20220901</startdate><enddate>20220901</enddate><creator>Stankov, Bogdan</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0003-1803-6629</orcidid></search><sort><creationdate>20220901</creationdate><title>Convergence Towards the End Space for Random Walks on Schreier Graphs</title><author>Stankov, Bogdan</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-73058f9edbdbee99992f533732b1006cb1492a0e394b21a30caed3e6a4540c5b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Convergence</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Probability Theory and Stochastic Processes</topic><topic>Random walk</topic><topic>Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Stankov, Bogdan</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of theoretical probability</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Stankov, Bogdan</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Convergence Towards the End Space for Random Walks on Schreier Graphs</atitle><jtitle>Journal of theoretical probability</jtitle><stitle>J Theor Probab</stitle><date>2022-09-01</date><risdate>2022</risdate><volume>35</volume><issue>3</issue><spage>1412</spage><epage>1422</epage><pages>1412-1422</pages><issn>0894-9840</issn><eissn>1572-9230</eissn><abstract>We consider a transitive action of a finitely generated group
G
and the Schreier graph
Γ
defined by this action for some fixed generating set. For a probability measure
μ
on
G
with a finite first moment, we show that if the induced random walk is transient, it converges towards the space of ends of
Γ
. As a corollary, we obtain that for a probability measure with a finite first moment on Thompson’s group
F
, the support of which generates
F
as a semigroup, the induced random walk on the dyadic numbers has a non-trivial Poisson boundary. Some assumption on the moment of the measure is necessary as follows from an example by Juschenko and Zheng.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10959-021-01104-6</doi><tpages>11</tpages><orcidid>https://orcid.org/0000-0003-1803-6629</orcidid></addata></record> |
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| ispartof | Journal of theoretical probability, 2022-09, Vol.35 (3), p.1412-1422 |
| issn | 0894-9840 1572-9230 |
| language | eng |
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| subjects | Convergence Mathematics Mathematics and Statistics Probability Theory and Stochastic Processes Random walk Statistics |
| title | Convergence Towards the End Space for Random Walks on Schreier Graphs |
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