In this paper we shall examine the relationship between weak convergence of processes S; 'ST,,, where B is Brownian motion, and convergence of the time changes T,, and the scale functions S,. In Section 2, we show that if the T,'s are inverses of functionals A, (not necessarily additive) and A,(t) +' A,(t) for each P= B,, x E R, then the processes ST,, converge weakly to ST,.
In Section 3, we consider linear regular diffusions in canonical form on C[O, co) and present necessary and sufficient conditions for weak convergence of these diffusions in terms of pathwise convergence of their time changes, weak convergence of their speed measures and pointwise convergence of their scale functions. The main theorem in this section is the following. If (X,):?, is a sequence of diffusions in canonical form, then (X,) converges weakly to X0 if and only if a.s. the diffusion paths converge uniformly on compact time intervals.
The general convergence theorem in Section 2, where the processes may not be diffusions, was motivated by approximating general diffusions by simpler processes [ 21.
1. PRELIMINARIES
For definitions and results concerning diffusions consult 14-6). Q = C[O, ao) will be given the topology of uniform convergence on compact sets; the u-algebras are those induced by the coordinate processes. All scale functions will map the reals iR onto the reals. B, denotes Weiner measure on .5(Q) starting at x. A set of full measure fl, is a set such that B,(QO) = 1 for every x. A functional (not necessarily additive) is a mapping A from [0, co) x a onto [0, co) such that there exists a set of full measure Q, such that A(0, w) = 0 and A(., w) is continuous and strictly increasing for w E Q,. T will denote the inverse of A. 200