A central limit theorem for extreme sojourns of diffusion processes

Abstract

Let X(t),t≥0, be a real-valued diffusion process having a stationary probability measure. For an increasing function u(t),(s) > u(t.) It is .shown, under general conditions on the diffusion coefficients, that if at a sufficiently slow rate, then has, for , a limiting normal distribution. The rate of increaseof u(t) is stated in terms of the scalefunction S(x) associated with the generator of the process; u(t) must satisfy , for . This complements an earlier result (Berman, 1988) in the case S(u(t)−t.where it was shown that there is a function vet) such that v(t)L(t) has a particular limiting compound Poisson distribution