Contiguity (probability theory)

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 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.

Let $(\Omega _{n},{\mathcal {F}}_{n})$ $(\Omega _{n},{\mathcal {F}}_{n})$ be a sequence of measurable spaces, each equipped with two measures P_n and Q_n.

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: Q_n is contiguous with respect to P_n if the “limiting support” of Q_n is a subset of the limiting support of P_n.

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