## The Annals of Probability

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

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

#### 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

**Digital Object Identifier**

doi:10.1214/aop/1176993227

**Mathematical Reviews number (MathSciNet)**

MR744233

**Zentralblatt MATH identifier**

0544.60011

**JSTOR**

links.jstor.org

**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. Ann. Probab. 12 (1984), no. 3, 768--793. doi:10.1214/aop/1176993227. http://projecteuclid.org/euclid.aop/1176993227.