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 f n n \{{f₍\}}₍ is a sequence of functions converging to f which verify some integrability conditions and θ n 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.
Bardina et al. (Thu,) studied this question.