Stochastic Sub-Additivity Approach to the Conditional Large Deviation Principle

Abstract

Given two Polish spaces $A_X$ and $A_Y$, let $\rho : A_X \times A_Y \to \mathbb{R}^d$ be a bounded measurable function. Let $X = {X_n : n \geq 1}$ and $Y = {Y_n : n \geq 1}$ be two independent stationary processes on $A_X^{\infty}$ and $A_Y^{\infty}$, respectively. The article studies the large deviation principle (LDP) for $n^{-1} \sum_{k=1}^n \rho(X_k, Y_k)$, conditional on $X$. Based on a stochastic version of approximate subadditivity, it is shown that if Y satisfies certain mixing condition, then for almost all random realization $x$ of $X$, the laws of $n^{-1} \sum_{k=1}^n \rho(x_k, Y_k)$ satisfy the conditional LDP with a non-random convex rate funcion. Conditions for the rate function to be non-trivial (that is, not $0/\infty$ function) are also given.