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Sanov Property, Generalized I-Projection and a Conditional Limit Theorem

Imre Csiszar
The Annals of Probability
Vol. 12, No. 3 (Aug., 1984), pp. 768-793
Stable URL: http://www.jstor.org/stable/2243326
Page Count: 26
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Sanov Property, Generalized I-Projection and a Conditional Limit Theorem
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Abstract

Known results on the asymptotic behavior of the probability that the empirical distribution $\hat P_n$ of an i.i.d. sample X1, ⋯, Xn belongs to a given convex set Π of probability measures, and new results on that of the joint distribution of X1, ⋯, Xn under the condition $\hat P_n \in \Pi$ are obtained simultaneously, using an information-theoretic identity. The main theorem involves the concept of asymptotic quasi-independence introduced in the paper. In the particular case when $\hat P_n \in \Pi$ is the event that the sample mean of a V-valued statistic ψ is in a given convex subset of V, a locally convex topological vector space, the limiting conditional distribution of (either) Xi is characterized as a member of the exponential family determined by ψ through the unconditional distribution PX, while X1, ⋯, Xn are conditionally asymptotically quasi-independent.

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