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

Heinz Cremers and Dieter Kadelka

Stochastic Processes and their Applications, 1986, vol. 21, issue 2, 305-317

Abstract: 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.

Date: 1986
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