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

Suppose Y1,Y2,…⊂Rd is a sequence of random variables such that the probability law of Yn/n satisfies the large deviation principle and suppose A⊂Rd. Let T(A)=inf{n: Yn∈A} be the first passage time and, to obtain a suitable scaling, let Tε(A)=εinf{n: Yn∈A/ε}. We consider the asymptotic behavior of Tε(A) as ε→0. We show that the the probability law of Tε(A) satisfies the large deviation principle; in particular, P{Tε(A)∈C}≈exp{−infτ∈CIA(τ)/ε} as ε→0, where IA(·) 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 Tε(A) and normalized first passage place YεTε(A).