Least squares estimator of fractional Ornstein Uhlenbeck processes with periodic mean

We first study the drift parameter estimation of the fractional Ornstein-Uhlenbeck process (fOU) with periodic mean for every $\frac{1}{2}<H<1$. More precisely, we extend the consistency proved in \cite{DFW} for $\frac{1}{2}<H<\frac{3}{4}$ to the strong consistency for any $\frac{1}{2}<H<1$ on the one hand, and on the other, we also discuss the asymptotic normality given in \cite{DFW}. In the second main part of the paper, we study the strong consistency and the asymptotic normality of the fOU of the second kind with periodic mean for any $\frac{1}{2}<H<1$.