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In [[probability theory]], two sequences of [[probability measure]]s are said to be '''contiguous''' if asymptotically they share the same [[support (measure theory)|support]]. Thus the notion of '''contiguity''' extends the concept of [[absolute continuity]] to the sequences of measures.

The concept was originally introduced by {{harvtxt|Le Cam|1960}} as part of his contribution to the development of abstract general [[asymptotic theory]] in mathematical [[statistics]]. Le Cam was instrumental during the period in the development of abstract general asymptotic theory in mathematical statistics. He is best known for the general concepts of [[local asymptotic normality]] and contiguity. <ref>Wolfowitz J.(1974) Review of the book: "Contiguity of Probability Measures: Some Applications in Statistics. by George G. Roussas",
[[Journal of the American Statistical Association]], 69, 278–279 [http://www.jstor.org/pss/2285551 jstor]</ref>

==Definition==
Let <math style="height:1.2em;position:relative;top:-.2em">(\Omega_n,\mathcal{F}_n)</math> be a sequence of [[measurable space]]s, each equipped with two measures ''Pn'' and ''Qn''.
* We say that ''Qn'' is '''contiguous''' with respect to ''Pn'' (denoted {{nowrap|''Qn'' ◁ ''Pn''}}) if for every sequence ''An'' of [[measurable set]]s, {{nowrap|''Pn''(''An'') → 0}} implies {{nowrap|''Qn''(''An'') → 0}}.
* The sequences ''Pn'' and ''Qn'' are said to be '''mutually contiguous''' or '''bi-contiguous''' (denoted {{nowrap|''Qn'' ◁▷ ''Pn''}}) if both ''Qn'' is contiguous with respect to ''Pn'' and ''Pn'' is contiguous with respect to ''Qn''. <ref>{{harvtxt|van der Vaart|1998|page=87}}</ref>

The notion of contiguity is closely related to that of [[Absolute continuity#Absolute continuity of measures|absolute continuity]]. We say that a measure ''Q'' is ''absolutely continuous'' with respect to ''P'' (denoted {{nowrap|''Q'' ≪ ''P''}}) if for any measurable set ''A'', {{nowrap|1 = ''P''(''A'') = 0}} implies {{nowrap|1 = ''Q''(''A'') = 0}}. That is, ''Q'' is absolutely continuous with respect to ''P'' if the [[support (measure theory)|support]] of ''Q'' is a subset of the support of ''P''. The ''contiguity'' property replaces this requirement with an asymptotic one: ''Qn'' is contiguous with respect to ''Pn'' if the “limiting support” of ''Qn'' is a subset of the limiting support of ''Pn''.

It is possible however that each of the measures ''Qn'' be absolutely continuous with respect to ''Pn'', while the sequence ''Qn'' not being contiguous with respect to ''Pn''.

The fundamental [[Radon–Nikodym theorem]] for absolutely continuous measures states that if ''Q'' is absolutely continuous with respect to ''P'', then ''Q'' has ''density'' with respect to ''P'', denoted as {{nowrap|1=''ƒ'' = {{frac|d''Q''|d''P''}}}}, such that for any measurable set ''A''
: <math>
Q(A) = \int_A f\,\mathrm{d}P, \,
</math>
which is interpreted as being able to “reconstruct” the measure ''Q'' from knowing the measure ''P'' and the derivative ''ƒ''. A similar result exists for contiguous sequences of measures, and is given by the ''Le Cam’s third lemma''.

==Applications==
*[[Econometrics]] <ref>http://www.samsi.info/200506/fmse/course-info/werker-updated-nov14.pdf</ref>

==See also==
*[[Contiguity]]
*[[Probability space]]

==Notes==
{{Reflist}}

==References==
{{refbegin}}
* {{cite book
| author = Hájek, J.
| coauthor = Šidák, Z.
| title = Theory of rank tests
| year = 1967
| publisher = Academic Press
| location = New York
| ref = CITEREFHájekŠidák1967
}}
* {{cite journal
| first = Lucien
| last = Le Cam
| authorlink = Lucien Le Cam
| year = 1960
| title = Locally asymptotically normal families of distributions
| journal = University of California Publications in Statistics
| volume = 3
| pages = 37–98
| ref = CITEREFLe_Cam1960
}}
* {{SpringerEOM
| last = Roussas
| first = George G.
| title = Contiguity of probability measures
| id = C/c120210
}}
* {{cite book
| last = van der Vaart
| first = A. W.
| title = Asymptotic statistics
| year = 1998
| publisher = Cambridge University Press
| ref = CITEREFvan_der_Vaart1998
}}
{{refend}}

==Additional literature==
:*Roussas, George G. (1972), ''Contiguity of Probability Measures: Some Applications in Statistics'', CUP, ISBN 978-0-521-09095-7.
:*Scott, D.J. (1982) Contiguity of Probability Measures, ''Australian & New Zealand Journal of Statistics'', 24 (1), 80–88.

==External references==
*[http://www.stat.yale.edu/~pollard/Books/Asymptopia/old-Contiguity.pdf Contiguity Asymptopia: 17 October 2000, David Pollard]
*[http://www.springerlink.com/content/h4v96n325l3627r7/ Asymptotic normality under contiguity in a dependence case ]
*[http://www.projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.aos/1176343289&page=record A Central Limit Theorem under Contiguous Alternatives]
*[http://www.stat.lsa.umich.edu/~moulib/stat612-notes3.pdf Superefficiency, Contiguity, LAN, Regularity, Convolution Theorems]
*[http://books.google.com/books?id=IlJE_9_e8UEC&pg=PA493&lpg=PA493&dq=Contiguity+space+asymptotic+normality&source=bl&ots=PAaWWTDYs6&sig=OoQ02c_PCTpaj3olElV6M-TV8qM&hl=en&ei=aohrSsqODJCiswOQs5iXBQ&sa=X&oi=book_result&ct=result&resnum=6 Testing statistical hypotheses]
*[http://www.iop.org/EJ/abstract/0036-0279/37/6/R06 Necessary and sufficient conditions for contiguity and entire asymptotic separation of probability measures R Sh Liptser et al 1982 Russ. Math. Surv. 37 107–136]
*[http://books.google.com/books?id=ifUA5CBiQ44C&pg=PA324&lpg=PA324&dq=%22Contiguity+space%22&source=bl&ots=6sgYRZGaz7&sig=Fd7FL3vexvpesM7NXS15qKcUJOk&hl=en&ei=6YBrSsT3D5TUsQOItpmXBQ&sa=X&oi=book_result&ct=result&resnum=1 The unconscious as infinite sets By Ignacio Matte Blanco, Eric (FRW) Rayner]
*[http://www3.interscience.wiley.com/journal/119856597/abstract?CRETRY=1&SRETRY=0 "Contiguity of Probability Measures", David J. Scott, La Trobe University]
*[http://www.jstor.org/pss/2242899 "On the Concept of Contiguity", Hall, Loynes]

[[Category:Probability theory]]

Almost simple group - Revision history

147.188.55.149: meaningless distinction removed