On the weak convergence of multiparameter stochastic integrals

Abstract In this paper we provide sufficient conditions for sequences of stochastic processes of the form ∫ [0, t ] f n ( u ) θ n ( u )d u , to weakly converge, in the space of continuous functions over a closed interval, to integrals with respect to the Brownian motion, ∫ [0, t ] f ( u ) W (d u ), where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> ${\left\{{f}_{n}\right\}}_{n}$ is a sequence of functions converging to f which verify some integrability conditions and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> ${\left\{{\theta }_{n}\right\}}_{n}$ is a sequence of stochastic processes whose integrals ∫ [0, t ] θ n ( u )d u converge in law to the Brownian motion (in the sense of the finite dimensional distribution convergence), in the multiparameter case.