  In the paper we consider the problem of estimating parameters entering the
drift of a fractional Ornstein-Uhlenbeck type process in the non-ergodic case,
when the underlying stochastic integral is of Young type. We consider the
sampling scheme that the process is observed continuously on $[0,T]$ and
$T\to\infty$. For known Hurst parameter $H\in(0.5, 1)$, i.e. the long range
dependent case, we construct a least-squares type estimator and establish
strong consistency. Furthermore, we prove a second order limit theorem which
provides asymptotic normality for the parameters of the periodic function with
a rate depending on $H$ and a non-central Cauchy limit result for the mean
reverting parameter with exponential rate. For the special case that the
periodicity parameter is the weight of a periodic function, which integrates to
zero over the period, we can even improve the rate to $\sqrt{T}$.
