A recent result of Takács (1995) gives explicitly the density of the time spent before t above a level x ≠ 0 by Brownian motion with drift. Takács' proof is by means of random walk approximations to Brownian motion, but in this paper we give two different proofs of this result by considerations involving only Brownian motion. We also give a reformulation of Takács' result which involves Brownian meanders, and an extension of Denisov's representation of Brownian motion in terms of two independent Brownian meanders.

An occupation measure describes the expected amount of time a stochastic process spends in different parts of its state space prior to a given random time. It is shown that a basic identity involving occupation measures provides a unified approach to a variety of moment identities for Markov chains, and some connections with potential theory are made.

The techniques used by Doeblin and Chung to obtain ordinary limit laws (central limit laws, weak and strong laws of large numbers, and laws of the iterated logarithm) for Markov chains, are extended to obtain analogous functional limit laws for stochastic processes which have embedded processes satisfying these laws. More generally, it is shown how functional limit laws of a stochastic process are related to those of a process embedded in it. The results herein unify and extend many existing limit laws for Markov, semi-Markov, queueing, regenerative, semi-stationary, and subordinated processes.