Limit Theorems of Occupation Times for Markov Processes

where u(f) is some normalizing function, has been investigated by many authors. A most general limit theorem was obtained by Darling and Kac [1], who also showed that, under suitable condition, the limit distribution must be Mittag-Leffler distribution. However, they confined themselves to the case where /(x) is nonnegative. C. Stone [5] derived a limit theorem for processes including 1 -dimensional diffusion processes with the infinitesimal generator -= -= — . It is not assumed that f ( x ) dm ax C is nonnegative, but it is essential that /(x) is not null-charged; \f(x)m(dx) *0. In this paper we study the case where f ( x ) is null-charged. If Xt is positively recurrent, the problem above can be reduced to the central limit theorem (see Tanaka [6]). A similar problem was treated by Dobrusin [2], who studied limit theorems for the 1 -dimensional simple random walk. The aim of this paper is to give a limit theorem for most general processes. Contrary to the case of [1], the limiting distribution is