Approximations of a complex Brownian motion with processes constructed from a process with independent increments

Abstract: In this paper, we show an approximation in law of the complex Brownian motion by processes constructed from a stochastic process with independent increments. We give sufficient conditions for the characteristic function of the process with independent increments that ensure the existence of the approximation. We apply these results to L\'evy processes. Finally we extend this results to the $m$-dimensional complex Brownian motion.