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 Lucien Le Cam in the 1960s 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.

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Definition

Let (Ω_n, F_n) be a sequence of measurable spaces, each equipped with two probability measures P_n and Q_n. Then Q_n is said to be contiguous with respect to P_n (denoted Q_n ◁ P_n) if P_n(A_n) → 0 implies Q_n(A_n) → 0 for every sequence of measurable sets A_n. The sequences P_n and Q_n are said to be mutually contiguous or bi-contiguous (denoted Q_n ◁▷ P_n) if both Q_n is contiguous with respect to P_n and P_n is contiguous with respect to Q_n.

Applications

• Econometrics

See also

• Contiguity
• Probability space

External References

• Contiguity Asymptopia: 17 October 2000, David Pollard
• Asymptotic normality under contiguity in a dependence case
• A Central Limit Theorem under Contiguous Alternatives
• Superefficiency, Contiguity, LAN, Regularity, Convolution Theorems
• Testing statistical hypotheses
• 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
• The unconscious as infinite sets By Ignacio Matte Blanco, Eric (FRW) Rayner
• "Contiguity of Probability Measures", David J. Scott, La Trobe University
• "On the Concept of Contiguity", Hall, Loynes

References

• van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press. {{cite book}}: Cite has empty unknown parameter: |1= (help)CS1 maint: ref duplicates default (link)