Exchangeable random variables: Difference between revisions

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Revision as of 02:07, 7 November 2018

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* Chow, Yuan Shih and Teicher, Henry, ''Probability theory. Independence, interchangeability, martingales,'' Springer Texts in Statistics, 3rd ed., Springer, New York, 1997. xxii+488 pp.&nbsp;{{ISBN|0-387-98228-0}}</ref><ref>Aldous, David J., ''Exchangeability and related topics'', in: École d'Été de Probabilités de Saint-Flour XIII — 1983, Lecture Notes in Math. 1117, pp.&nbsp;1–198, Springer, Berlin, 1985. {{ISBN|978-3-540-15203-3}} {{doi|10.1007/BFb0099421}}</ref>

(A sequence ''E''₁, ''E''₂, ''E''₃, ... of events is said to be exchangeable precisely if the sequence of its [[indicator function]]s is exchangeable.) The distribution function ''F''_{''X''_''1'',...,''X''_''n''}(''x''₁, ..., ''x''_''n'') of a finite sequence of exchangeable random variables is symmetric in its arguments {{nowrap|''x''₁, ... ,''x''_''n''.}} [[Olav Kallenberg]] provided an appropriate definition of exchangeability for continuous-time stochastic processes.<ref>{{cite journal | last=Diaconis| first=Persi | authorlink=Persi Diaconis | title=Book review: ''Probabilistic symmetries and invariance principles'' (Olav Kallenberg, Springer, New York, 2005) | journal=Bulletin of the AmerericanAmerican Mathematical Society (|series=New Series) | volume=46 | year=2009 | issue=4 | doi=10.1090/S0273-0979-09-01262-2 | url=http://www.ams.org/journals/bull/2009-46-04/S0273-0979-09-01262-2/home.html | pages=691–696 | mr=2525743}}</ref><ref name="Kallenberg" >[[Olav Kallenberg|Kallenberg, O.]], ''Probabilistic symmetries and invariance principles''. Springer-Verlag, New York (2005). 510 pp.&nbsp;{{ISBN|0-387-25115-4}}.</ref>

== History ==

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* {{cite book |last=Borovskikh | first=Yu. V. | title=''U''-statistics in Banach spaces | publisher=VSP | ___location=Utrecht | year=1996 | pages=xii+420 | isbn=90-6764-200-2 | mr=1419498 }}

* Chow, Yuan Shih and Teicher, Henry, ''Probability theory. Independence, interchangeability, martingales,'' Springer Texts in Statistics, 3rd ed., Springer, New York, 1997. xxii+488 pp.&nbsp;{{ISBN|0-387-98228-0}}

* {{cite journal | last=Diaconis| first=Persi | authorlink=Persi Diaconis | title=Book review: ''Probabilistic symmetries and invariance principles'' (Olav Kallenberg, Springer, New York, 2005) | journal=Bulletin of the AmerericanAmerican Mathematical Society (|series=New Series) | volume=46 | year=2009 | issue=4 | doi=10.1090/S0273-0979-09-01262-2 | url=http://www.ams.org/journals/bull/2009-46-04/S0273-0979-09-01262-2/home.html | pages=691–696 | mr=2525743}}

* [[Olav Kallenberg|Kallenberg, O.]], ''Probabilistic symmetries and invariance principles''. Springer-Verlag, New York (2005). 510 pp.&nbsp;{{ISBN|0-387-25115-4}}.

* Kingman, J. F. C., ''Uses of exchangeability'', Ann. Probability 6 (1978) 83&ndash;197 {{MR|494344}} {{JSTOR|2243211}}