A General Conditional Large Deviation Principle

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.

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