A General Conditional Large Deviation Principle

TL;DR

This paper establishes a general large deviation principle for conditioned measures on Hausdorff spaces, extending the classical theory by identifying conditions under which the conditioned measures inherit a large deviation principle with an explicitly defined rate function.

Contribution

It provides a broad, applicable framework for understanding large deviations of conditioned measures, including explicit conditions and a formula for the new rate function.

Findings

Conditional measures satisfy a large deviation principle under specified conditions.

The rate function for conditioned measures is explicitly derived from the original rate function.

The results extend classical large deviation theory to more general settings.

Abstract

Given a sequence of Borel probability measures on a Hausdorff space which satisfy a large deviation principle, we consider the corresponding sequence of measures formed by conditioning on a set $B$. If the large deviation rate function $I$ is good and effectively continuous and the conditioning set has the property that (1) $\overline{B^\circ} = \overline{B}$ and (2) $I(x) < \infty$ for all $x \in \overline{B}$, then the sequence of conditional measures satisfies a large deviation principle with the good, effectively continuous rate function $I_B$, where $I_B(x) = I(x)-\inf I(B)$ if $x\in\overline{B}$ and $I_B(x) = \infty$ otherwise.