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

Suppose Y-1, Y-2,... subset of R-d is a sequence of random variables such that the probability law of Y-n/n satisfres the large deviation principle and suppose A subset of R-d. Let T(A)= inf{n: Y-n is an element of A} be the first passage time and, to obtain a suitable scaling, let T-epsilon(A) = epsilon inf{n: Y-n is an element of A/epsilon}. We consider the asymptotic behavior of T-epsilon(A) as epsilon --> 0. We show that the the probability law of T-epsilon(A) satisfies the large deviation principle; in particular, P{T-epsilon(A) is an element of C} approximate to exp{- inf(tau is an element of C)I(A)(tau)/epsilon} as epsilon --> 0, where I-A(.) 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-epsilon(A) and normalized first passage place Y-T epsilon(A)(epsilon). (C) 1998 Elsevier Science B.V. All rights reserved.