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.
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The concept was originally introduced by 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.
Definition
Let
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 Q ≪ P) if for any measurable set A, P(A) = 0 implies Q(A) = 0. 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 ƒ = dQ⁄dP, 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.