  First, sufficient conditions are given for a triangular array of random
vectors such that the sequence of related random step functions converges
towards a (not necessarily time homogeneous) diffusion process. These
conditions are weaker and easier to check than the existing ones in the
literature, and they are derived from a very general semimartingale convergence
theorem due to Jacod and Shiryaev, which is hard to use directly. Next,
sufficient conditions are given for convergence of stochastic integrals of
random step functions, where the integrands are functionals of the integrators.
This result covers situations which can not be handled by existing ones.
