LetX(t),t≥0, be a real-valued diffusion process having a stationary probability measure. For an increasing functionu(t),(s) >u(t.) It is .shown, under general conditions on the diffusion coefficients, that ifat a sufficiently slow rate, thenhas, for, a limiting normal distribution. The rate of increaseofu(t) is stated in terms of the scalefunctionS(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 thatv(t)L(t) has a particular limiting compound Poisson distribution