Stochastic Sub - Additivity Approach to the Conditional Large Deviation Principle

University of Chicago Given two Polish spaces AX and AY, let ρ AX × AY → d be a bounded measurable function. Let X = Xn n ≥ 1 and Y = Yn n ≥ 1 be two independent stationary processes on AX and A ∞ Y , respectively. The article studies the large deviation principle (LDP) for n−1 ∑n k=1 ρ Xk Yk , 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 ∑n k=1 ρ xk Yk satisfy the conditional LDP with a non-random convex rate function. Conditions for the rate function to be non-trivial (that is, not 0/∞ function) are also given.