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We consider a mixed moving average (MMA) process X driven by a Lévy basis and prove that it is weakly dependent with rates computable in terms of the moving average kernel and the characteristic quadruple of the Lévy 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.An important example of MMA are the supOU processes studied in [3,10,25,27]. In the univariate case, assume |x|>1 log(|x|) ν(dx) < ∞ and R − − 1 A π(dA) < ∞, where ν is a Lévy measure and π is the probability distribution on R − of the random parameter A, see Definition 2.1 for details. If Λ is a Lévy basis on R with those characteristics, then the processis called a supOU process. Whereas a non-Gaussian Ornstein-Uhlenbeck process necessarily exhibits autocorrelation e ah for h ∈ N, the supOU process has a flexible dependence structure. For example, its autocorrelations can show a polynomial decay depending on the probability distribution π. Moreover, when a discrete probability distribution π for the random parameter A is considered, we obtain a popular model used, for example, in stochastic volatility models [3], in modeling fractal activity times [36,37] and in astrophysics [35].MMA processes can also be used, under suitable conditions, as building blocks for more complex models. We study in this paper the class of MMA stochastic volatility models. An example of the class is the supOU SV model, defined in [10,11], where the log-price process (of some financial asset) is defined for t ∈ R + as

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We consider a Lévy driven continuous time moving average process X sampled at random times which follow a renewal structure independent of X. Asymptotic normality of the sample mean, the sample autocovariance, and the sample autocorrelation is established under certain conditions on the kernel and the random times. We compare our results to a classical non-random equidistant sampling method and give an application to parameter estimation of the Lévy driven Ornstein-Uhlenbeck process.Mathematics subject classification: 60F05, 60G10, 62D05.