Continuous Auto-Regressive Moving Average Random Fields on Rn

Summary We define an isotropic Lévy-driven continuous auto-regressive moving average CARMA(p, q) random field on Rn 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 Rn 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 Rn. 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.