  In the present paper we study the asymptotic behavior of the auto-covariance
function for Ornstein-Uhlenbeck (OU) processes driven by Gaussian noises with
stationary and non-stationary increments and for Hermite OU processes. Our
results are generalizations of the corresponding results of Cheridito et al.
\cite{CKM} and Kaarakka and Salminen \cite{KS}.
