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In probability theory, two sequences of probability measures are said to be contiguous if asymptotically they share the same support. Thus the notion of contiguity extends the concept of absolute continuity to the sequences of measures. The concept was originally introduced by 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.〔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 (jstor )〕 ==Definition== Let be a sequence of measurable spaces, each equipped with two measures ''Pn'' and ''Qn''. * We say that ''Qn'' is contiguous with respect to ''Pn'' (denoted ) if for every sequence ''An'' of measurable sets, implies . * The sequences ''Pn'' and ''Qn'' are said to be mutually contiguous or bi-contiguous (denoted ) if both ''Qn'' is contiguous with respect to ''Pn'' and ''Pn'' is contiguous with respect to ''Qn''. The notion of contiguity is closely related to that of absolute continuity. We say that a measure ''Q'' is ''absolutely continuous'' with respect to ''P'' (denoted ) if for any measurable set ''A'', implies . That is, ''Q'' is absolutely continuous with respect to ''P'' if the 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 , such that for any measurable set ''A'' : 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''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Contiguity (probability theory)」の詳細全文を読む スポンサード リンク
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