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Summary We define an isotropic Lévy‐driven continuous auto‐regressive moving average CARMA(p,q) random field on double-struckRn as the integral of a radial CARMA kernel with respect to a Lévy sheet. Such fields constitute a parametric family characterized by an auto‐regressive polynomial a and a moving average polynomial b having zeros in both the left and the right complex half‐planes. They extend the well‐balanced Ornstein–Uhlenbeck process of Schnurr and Woerner to a well‐balanced CARMA process in one dimension (with a much richer class of autocovariance functions) and to an isotropic CARMA random field on double-struckRn for n>1. We derive second‐order properties of these random fields and extend the results to a larger class of anisotropic CARMA random fields. If the driving Lévy sheet is compound Poisson it is trivial to simulate the corresponding random field on any bounded subset of double-struckRn. A method for joint estimation of the CARMA kernel parameters and knot locations is proposed for compound‐Poisson‐driven fields and is illustrated by applications to simulated data and Tokyo land price data.

The purpose of the paper is to propose a frequency domain approach for irregularly spaced data on R d . We extend the original definition of a periodogram for time series to that for irregularly spaced data and define non-parametric and parametric spectral density estimators in a way that is similar to the classical approach. Introduction of the mixed asymptotics, which are one of the asymptotics for irregularly spaced data, makes it possible to provide asymptotic theories to the spectral estimators.The asymptotic result for the parametric estimator is regarded as a natural extension of the classical result for regularly spaced data to that for irregularly spaced data. Empirical studies are also included to illustrate the frequency domain approach in comparisons with the existing spatial and frequency domain approaches.

We propose a test for separability of the correlation structure of a multivariate time series. We construct test statistics based on a spectral density matrix estimated in a nonparametric way and derive their asymptotic properties. We use simulation to check the performance in finite samples.