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Published online by Cambridge University Press: 04 February 2011
Summary
The main purpose of this chapter is to present the concept of contiguity (see Definition 2.1) introduced by LeCam [4] and study some alternative characterizations of it (see Theorem 6.1). In the process of doing so, some auxiliary concepts such as weak convergence, relative compactness and tightness of a sequence of probability measures are needed. These concepts are introduced in this chapter, as we go along, and also some of their relationships are stated and/or proved. For the omitted proofs, the reader is always referred to appropriate sources. The various characterizations of continguity provide alternative methods one may employ in establishing the presence (or absence) of contiguity in a given case. Some concrete examples are used for illustrative purposes.
Contiguity is a concept of ‘nearness’ of sequences of probability measures. It would then be appropriate to relate it to other more familiar concepts of the same nature such as ‘nearness’ of two sequences of probability measures expressed by the norm (L1-norm) associated with convergence in variation. By means of examples, it is shown, as one would expect, that ‘nearness’ of two sequences of probability measures expressed by contiguity is weaker than that expressed by the L1-norm.
Some attention is also focused to possible relationships between contiguity on the one hand, and mutual absolute continuity and tightness on the other. In connection with this, it is shown, by means of examples, that mutual absolute continuity of the (corresponding) measures in two sequences of probability measures need not imply contiguity of the sequences.
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