Exchangeable random variables: Difference between revisions

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Revision as of 15:28, 7 December 2020

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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 | authorlinkauthor-link=Persi Diaconis | title=Book review: ''Probabilistic symmetries and invariance principles'' (Olav Kallenberg, Springer, New York, 2005) | journal=Bulletin of the American Mathematical Society |series=New Series | volume=46 | year=2009 | issue=4 | doi=10.1090/S0273-0979-09-01262-2 | pages=691–696 | mr=2525743| doi-access=free }}</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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We can then solve the inequality for the covariance yielding the stated lower bound. The non-negativity of the covariance for the infinite sequence can then be obtained as a limiting result from this finite sequence result.

Equality of the lower bound for finite sequences is achieved in a simple urn model: An urn contains 1 red marble and ''n''&nbsp;&minus;&nbsp;1 green marbles, and these are sampled without replacement until the urn is empty. Let ''X''_''i''&nbsp;=&nbsp;1 if the red marble is drawn on the ''i''-th trial and 0 otherwise. A finite sequence that achieves the lower covariance bound cannot be extended to a longer exchangeable sequence.<ref>{{cite book|title=Limit theorems for sums of exchangeable random&nbsp;variables|first1=Robert Lee|last1=Taylor|first2=Peter Z.|last2=Daffer|first3=Ronald F.|last3=Patterson|mr=|ref=harv|publisher=Rowman and Allanheld|url=https://books.google.com/books?id=6RaoAAAAIAAJ|year=1985|pages=1–152|isbn=9780847674350}}</ref>

== Examples ==

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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 | authorlinkauthor-link=Persi Diaconis | title=Book review: ''Probabilistic symmetries and invariance principles'' (Olav Kallenberg, Springer, New York, 2005) | journal=Bulletin of the American Mathematical Society |series=New Series | volume=46 | year=2009 | issue=4 | doi=10.1090/S0273-0979-09-01262-2 | pages=691–696 | mr=2525743| doi-access=free }}

* [[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}}

* O'Neill, B. (2009) Exchangeability, Correlation and Bayes' Effect. ''International Statistical Review'' '''77(2)''', pp.&nbsp;241–250. {{ISBN|978-3-540-15203-3}} {{doi|10.1111/j.1751-5823.2008.00059.x}}

* {{cite book|title=Limit theorems for sums of exchangeable random&nbsp;variables|first1=Robert Lee|last1=Taylor|first2=Peter Z.|last2=Daffer|first3=Ronald F.|last3=Patterson|mr=|ref=harv|publisher=Rowman and Allanheld|url=https://books.google.com/books?id=6RaoAAAAIAAJ|year=1985|pages=1–152|isbn=9780847674350}}

* Zabell, S. L. (1988) "Symmetry and its discontents", in Skyrms, B. & Harper, W. L. ''Causation, Chance and Credence, ''pp''155-190, Kluwer

* {{ cite journal | author=&mdash; | title=Predicting the unpredictable | year=1992 | journal=Synthese | volume=90 | issue=2 | page=205 | doi=10.1007/bf00485351}}