## The Annals of Probability

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

Imre Csiszar

#### 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

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
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.