On weak convergence of stochastic processes with Lusin path spaces (Q788370)

The perhaps most widely used concept for proving weak convergence of stochastic processes with paths in a topological function space [X], due to Prokhorov, requires (i) the verification of weak convergence of the finite dimensional distributions, (ii) the validation of a tightness condition, and (iii) the proof that (i) and (ii) in fact imply weak convergence of the processes. The main result of the present paper is that, for a large class of path spaces, (iii) is always satisfied: Let (T,\({\mathcal B},\nu)\) be a \(\sigma\)-finite measure space and [X] be a set of real valued measurable functions on T. As usual functions are identified if they are \(\nu\)-almost equal. Now (iii) holds true, if [X] is endowed with a Lusin topology which is finer than that of convergence in \(\nu\)-measure. Demonstrably such spaces are C(T), D[0,1], \(L_ p(\nu)\), and Lipschitz function spaces. Generalizations to Banach space valued stochastic processes are possible and will appear elsewhere.