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August, 1984 Sanov Property, Generalized $I$-Projection and a Conditional Limit Theorem
Imre Csiszar
Ann. Probab. 12(3): 768-793 (August, 1984). DOI: 10.1214/aop/1176993227


Known results on the asymptotic behavior of the probability that the empirical distribution $\hat P_n$ of an i.i.d. sample $X_1, \cdots, X_n$ belongs to a given convex set $\Pi$ of probability measures, and new results on that of the joint distribution of $X_1, \cdots, X_n$ 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 $\psi$ is in a given convex subset of $V$, a locally convex topological vector space, the limiting conditional distribution of (either) $X_i$ is characterized as a member of the exponential family determined by $\psi$ through the unconditional distribution $P_X$, while $X_1, \cdots, X_n$ are conditionally asymptotically quasi-independent.


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Imre Csiszar. "Sanov Property, Generalized $I$-Projection and a Conditional Limit Theorem." Ann. Probab. 12 (3) 768 - 793, August, 1984.


Published: August, 1984
First available in Project Euclid: 19 April 2007

zbMATH: 0544.60011
MathSciNet: MR744233
Digital Object Identifier: 10.1214/aop/1176993227

Primary: 60F10
Secondary: 60B10 , 62B10 , 82A05 , 94A17

Keywords: $I$-projection , asymptotic quasi-independence , exponential family , Kullback-Leibler information , large deviations in abstract space , maximum entropy principle

Rights: Copyright © 1984 Institute of Mathematical Statistics

Vol.12 • No. 3 • August, 1984
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