Large deviations of continuous regular conditional probabilities

TL;DR

This paper establishes conditions under which regular conditional probabilities satisfy large deviation principles, extending Sanov's theorem to conditioned empirical distributions in a product space setting.

Contribution

It provides necessary and sufficient conditions for large deviations of regular conditional probabilities and introduces a Sanov-type theorem for conditioned empirical distributions.

Findings

Conditions for large deviations of conditional probabilities are characterized.

A Sanov-type theorem for empirical distributions conditioned on another coordinate is derived.

Regular conditional probabilities can be viewed as a special case of product regular conditional probabilities.

Abstract

We study product regular conditional probabilities under measures of two coordinates with respect to the second coordinate that are weakly continuous on the support of the marginal of the second coordinate. Assuming that there exists a sequence of probability measures on the product space that satisfies a large deviation principle, we present necessary and sufficient conditions for the conditional probabilities under these measures to satisfy a large deviation principle. The arguments of these conditional probabilities are assumed to converge. A way to view regular conditional probabilities as a special case of product regular conditional probabilities is presented. This is used to derive conditions for large deviations of regular conditional probabilities. In addition, we derive a Sanov-type theorem for large deviations of the empirical distribution of the first coordinate conditioned on fixing the empirical distribution of the second coordinate.