On Convergence of Stochastic Processes

where £iX) is the distribution function of the random variable X,f( ) is a real-valued function 5 continuous almost everywhere (p), and the limit is in the sense of the usual weak convergence of distributions. Equation (2) is usually the real center of interest, for many " limit-distribution theorems" are implicit in it. It is clear that for given {pn} and p, the better theorem of this kind would be the one in which (2) is proved for the larger class of functions /.In this paper we shall show that certain known "invariance principles" can under some hypotheses be improved by considerably enlarging the class of functions for which (2) holds. This will be done by considering spaces S other than the customary ones. For example, in studying convergence to the Wiener process, it is usual to let S be the space (denoted 6) of continuous functions with the uniform topology. However, this choice does not fully exploit the pleasant properties of the Wiener path-functions, which are not only continuous but also Holder continuous of any order up to 1/2. Therefore we shall attempt to use spaces Lipa in place of 6 as the function-space 5. When weak convergence can be established using such spaces, the class of functionals for which (2) is known to hold becomes much larger than before. To carry out the idea sketched above it is necessary to have a criterion which guarantees that the sample functions of a stochastic process are a.s.