First passage times of general sequences of random vectors: A large deviations approach

Abstract

Suppose Y 1,Y 2,…⊂ R d is a sequence of random variables such that the probability law of Y n / n satisfies the large deviation principle and suppose A⊂ R d . Let T(A)= inf{n : Y n∈A} be the first passage time and, to obtain a suitable scaling, let T ε(A)=ε inf{n : Y n∈A/ε} . We consider the asymptotic behavior of T ε ( A) as ε→0. We show that the the probability law of T ε ( A) satisfies the large deviation principle; in particular, P { T ε ( A)∈ C}≈exp{−inf τ∈ C I A ( τ)/ ε} as ε→0, where I A (·) is a large deviation rate function and C is any open or closed subset of [0,∞). We then establish conditional laws of large numbers for the normalized first passage time T ε ( A) and normalized first passage place Y ε T ε ( A) .