  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 towards the time-changed Ornstein-Uhlenbeck as the Hurst
index $H \to 1/2^+$, with locally uniform convergence of one-dimensional
distributions. Moreover, we also achieve convergence in the Skorohod
$J_1$-topology of the time-changed fractional Ornstein-Uhlenbeck process as $H
\to 1/2^+$ in the space of c\`adl\`ag functions. Finally, we exploit some
convergence properties of mild solutions of a generalized Fokker-Planck
equation associated to the aforementioned processes, as $H \to 1/2^+$.
