Darling and Kac revisited

Abstract

The famous paper of Darling and Kac (1957) obtains the distribution of ∫¹₀f(X(u)) du for a wide class of nonnegative functions f and Markov processes X(.). Their hypothesis and methods do not permit extension to include more general functions f. In this note two different sets of hypothesis on X(.) are presented under which it is shown that for all f integrable w.r.t. an approximate measure π (.), there is a nondegenerate limit distribution for ∫¹₀f(X(u)) du when suitably normalized. This is then applied to Brownian motion in one and two dimensions, stable process and processes with independent increments.