On weak convergence of integral functionals of stochastic processes with applications to processes taking paths in LEP

The weak convergence of certain functionals of a sequence of stochastic processes is investigated. The functionals under consideration are of the form f[phi](x) = [integral operator] [phi] (t, x(t))[mu](dt). The main result is as follows: If a sequence is weakly tight in a certain sense, and, in addition, the finite dimensional distributions of the processes converge weakly, then this implies weak convergence of the functionals (f[phi]1([xi]n),..., f[phi]m([xi]n)) to (f[phi]1([xi]0),..., f[phi]m([xi]0)). Necessary and sufficient conditions for weak tightness are stated and applications of the results to the case of LEp-valued stochastic processes are given, ln particular it is shown that the usual tightness condition for weak convergence of such processes can be considerably weakened.