Dominating Points and the Asymptotics of Large Deviations for Random Walk on $\mathbb{R}^d$

Abstract

Let $\mu(\cdot)$ be a probability measure on $\mathbb{R}^d$ and $B$ be a convex set with nonempty interior. It is shown that there exists a unique "dominating" point associated with $(\mu, B)$. This fact leads (via conjugate distributions) to a representation formula from which sharp asymptotic estimates of the large deviation probabilities $\mu^{\ast n}(nB)$ can be derived.