  We consider a mixed moving average (MMA) process X driven by a L\'evy basis
and prove that it is weakly dependent with rates computable in terms of the
moving average kernel and the characteristic quadruple of the L\'evy basis.
Using this property, we show conditions ensuring that sample mean and
autocovariances of X have a limiting normal distribution. We extend these
results to stochastic volatility models and then investigate a Generalized
Method of Moments estimator for the supOU process and the supOU stochastic
volatility model after choosing a suitable distribution for the mean reversion
parameter. For these estimators, we analyze the asymptotic behavior in detail.
