Weak convergence of the complex fractional Brownian motion

Abstract

In this paper, we obtain two approximations in law of the complex fractional Brownian motion by processes constructed from a Poisson process and a Lévy process, respectively. MSC: 60F05; 60G15; 60G22

Weak approximation of the complex Brownian sheet from a Lévy sheet and applications to SPDEs

“…Hence, in view of the above theorem, we observe that the real and imaginary parts of X ε are clearly not independent, for any ε > 0, while in the limit they are. This phenomenon is not new, for it already appeared in the study of analogous problems in the one-parameter setting (see, e.g., [1,4,10]). Indeed, in [1], a family of processes that converges in law to a complex Brownian motion was constructed from a unique Poisson process.…”

“…Lévy processes are one of the examples where the latter results may be applied. Finally, the authors of [10] use Poisson and Lévy processes in order to obtain approximations in law of a complex fractional Brownian motion.…”