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

Suppose is a sequence of random variables such that the probability law of Yn/n satisfies the large deviation principle and suppose . Let T(A)=inf{n: Yn[set membership, variant]A} be the first passage time and, to obtain a suitable scaling, let T[var epsilon](A)=[var epsilon]inf{n: Yn[set membership, variant]A/[var epsilon]}. We consider the asymptotic behavior of T[var epsilon](A) as [var epsilon]-->0. We show that the the probability law of T[var epsilon](A) satisfies the large deviation principle; in particular, as [var epsilon]-->0, where IA(·) is a large deviation rate function and C is any open or closed subset of [0,[infinity]). We then establish conditional laws of large numbers for the normalized first passage time T[var epsilon](A) and normalized first passage place Y[var epsilon]T[var epsilon](A).