Convergence results for the time-changed fractional ornstein–uhlenbeck processes

In this paper we study some convergence results concerning the one-dimensional distribution of a time-changed fractional Ornstein–Uhlenbeck process. In particular, we establish that, despite the time change, the process admits a Gaussian limit random variable. On the other hand, we prove that the process converges to-wards the time-changed Ornstein–Uhlenbeck as the Hurst index H → 1/2+,with locally uniform convergence of one-dimensional distributions. Moreover, we also achieve convergence in the Skorokhod J1-topology of the time-changed fractional Ornstein–Uhlenbeck process as H → 1/2+ in the space of càdlàg functions. Finally, we exploit some convergence properties of mild solutions of a generalized Fokker– Planck equation associated to the aforementioned processes, as H → 1/2+.