Estimation of Lévy-driven CARMA models under renewal sampling

Continuous-time autoregressive and moving average (CARMA) models are extensively used to model high-frequency and irregularly sampled data. We study Whittle estimation for the model parameters when the process is observed at renewal times. The driving noise is assumed to be a Lévy process allowing for more flexibility including heavy-tailed marginal distributions and jumps in the sample paths. We show that the Whittle estimator based on the integrated periodogram is consistent and asymptotically normal under very mild conditions. To obtain these results, we establish the asymptotic normality of the integrated periodogram.

💡 Research Summary

This paper addresses the problem of parameter estimation for Continuous-time Autoregressive Moving Average (CARMA) models when the driving noise is a Lévy process and the observation times follow a renewal sampling scheme. CARMA models are widely used for modeling high-frequency and irregularly spaced data in fields such as finance, signal processing, and the natural sciences. The use of a Lévy process as the driving noise introduces significant flexibility, allowing for heavy-tailed marginal distributions, asymmetry, and jumps in the sample paths—features often observed in real-world data. Renewal sampling, where the inter-arrival times between observations are independent and identically distributed, realistically models scenarios like irregular financial trades or battery-saving modes in wearable devices, and crucially helps avoid the aliasing problem inherent in equidistant sampling.

The authors propose a Whittle-type estimation procedure based on the integrated periodogram of the sampled process. The core idea is to define an objective function K(θ) that is maximized at the true parameter value θ_0, where K(θ) integrates the log-ratio of the normalized spectral density of the sampled process. Since this spectral density is unknown, it is replaced by the integrated periodogram I_{Z,n}(u) calculated from the observed data, yielding an estimator \hat{K}_n(θ). The parameter estimator \hat{θ}_n is then defined as the maximizer of \hat{K}_n(θ) over a compact parameter space Θ.

The main theoretical contributions of the paper are twofold. First, the authors establish the asymptotic normality of the integrated periodogram under mild conditions. This proof leverages the fact that the sampled sequence (Y(τ_k), τ_k - τ_{k-1}) forms a strongly mixing process with exponentially decaying coefficients, a property inherited from the underlying CARMA process. Second, using this result, they prove the asymptotic normality of the parameter estimator itself. Specifically, they show that √n ( \hat{θ}_n - θ_0 ) converges in distribution to a multivariate normal distribution with mean zero and a covariance matrix Σ_0 = W^{-1} Q W^{-1}, where W is the Hessian matrix of the expected objective function K(θ) at θ_0, and Q is a covariance matrix derived from the asymptotic distribution of the integrated periodogram’s gradient. A significant improvement over prior work is that these results require only a finite moment of order 4+δ for the Lévy process, rather than moments of all orders.

Furthermore, the paper demonstrates that the same asymptotic distribution holds under two different asymptotic regimes: when the number of observations n tends to infinity, and when the observation time horizon T tends to infinity (with the number of observations N(T) growing proportionally). Consistency of the estimators follows directly from the asymptotic normality. The paper also outlines a consistent estimator for the variance σ_L^2 of the driving Lévy process.

The methodology is illustrated through the special case of an Ornstein-Uhlenbeck process (a CARMA(1,0) model). Simulation studies are conducted where the driving noise is either a Brownian motion or a centered Gamma process, with exponentially distributed inter-arrival times. The results show that the mean of the parameter estimates converges to the true value, and their variance decreases as the sample size increases, empirically validating the theoretical findings. The paper concludes by noting the advantages of the renewal sampling scheme in preventing aliasing and the flexibility gained from the Lévy-driven framework, suggesting avenues for future research.