  We study statistical inference of the drift parameters for the Volterra
Ornstein-Uhlenbeck process on R and the Volterra Cox-Ingersoll-Ross process on
R+ in the ergodic regime. For continuous-time observations, we derive the
corresponding maximum likelihood estimators and show that they are strongly
consistent and asymptotically normal locally uniformly in the parameters. For
the case of discrete high-frequency observations, we prove similar results by
discretization of the continuous-time maximum likelihood estimator. Finally,
for discrete low-frequency observations, we show that the method of moments is
consistent. Our proofs are crucially based on the law of large numbers. To
prove the latter, we introduce the notion of asymptotic independence which has
the advantage that it can be effectively verified by the affine transformation
formula and convergence of the characteristic function. As a side product of
our results, we show that the stationary processes are ergodic.
