The Annals of Probability

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

Imre Csiszar

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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 $X_1, \cdots, X_n$ 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.

Article information

Source
Ann. Probab. Volume 12, Number 3 (1984), 768-793.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176993227

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aop/1176993227

Mathematical Reviews number (MathSciNet)
MR744233

Zentralblatt MATH identifier
0544.60011

Subjects
Primary: 60F10: Large deviations
Secondary: 60B10: Convergence of probability measures 62B10: Information-theoretic topics [See also 94A17] 94A17: Measures of information, entropy 82A05

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

Citation

Csiszar, Imre. Sanov Property, Generalized $I$-Projection and a Conditional Limit Theorem. The Annals of Probability 12 (1984), no. 3, 768--793. doi:10.1214/aop/1176993227. http://projecteuclid.org/euclid.aop/1176993227.


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