Hitting probabilities and large deviations

Abstract

Let ${Y_n}_{n \in \mathbb{Z}_+}$ be a sequence of random variables in $\mathbb{R}^d$ and let $A \subset \mathbb{R}^d$. Then $\mathbf{P}\{Y_n \in A \text{for some $n$}\}$ is the hitting probability of the set A by the sequence ${Y_n}$. We consider the asymptotic behavior, as $m \to \infty$, of $\mathbf{P}\{Y_n \in mA \text{some $n$}\} = \mathbf{P}{\text{hitting $mA$}$ whenever (1) the probability law of $Y_n/n$ satisfies the large deviation principle and (2) the central tendency of $Y_n/n$ is directed away from the given set A. For a particular function $\tilde{I}$, we show $m \to \infty$, of $\mathbf{P}\{Y_n \in mA \text{some $n$}\} \approx \exp (-m \tilde{I}(A))$.