Bridging between short-range and long-range dependence with mixed spatio-temporal Ornstein–Uhlenbeck processes

Abstract

While short-range dependence is widely assumed in the literature for its simplicity, long-range dependence is a feature that has been observed in data from finance, hydrology, geophysics and economics. In this paper, we extend a Lévy-driven spatio-temporal Ornstein-Uhlenbeck process by randomly varying its rate parameter to model both short-range and longrange dependence. This particular set-up allows for non-separable spatio-temporal correlations which are desirable for real applications, as well as flexible spatial covariances which arise from the shapes of influence regions. Theoretical properties such as spatio-temporal stationarity and second-order moments are established. An isotropic g-class is also used to illustrate how the memory of the process is related to the probability distribution of the rate parameter. We develop a simulation algorithm for the compound Poisson case which can be used to approximate other Lévy bases. The generalised method of moments is used for inference and simulation experiments are conducted with a view towards asymptotic properties.

Central limit theorems for stationary random fields under weak dependence with application to ambit and mixed moving average fields

“…We apply the developed asymptotic theory to mixed spatio-temporal Ornstein-Uhlenbeck (MSTOU) processes. MSTOU processes were introduced in [47] and extend spatio-temporal Ornstein-Uhlenbeck (STOU) processes (see [11], [46]) by additionally mixing the mean reversion parameter. Moreover, this extension can cover short-range as well as long-range dependence structures in space-time.…”

“…Moreover, we exploit the hereditary properties of η-as well as θ-lex-weak dependence and obtain conditions for the sample moments of order p with p ≥ 1 to be asymptotic normally distributed. Finally, we give explicit computations for mixed spatio-temporal Ornstein-Uhlenbeck processes [47], also called MSTOU processes, and Lévy-driven CARMA fields [22,51]. In particular, our calculations in the case of the MSTOU processes show that it is possible to determine the asymptotic normality of the generalized method of moments estimator, GMM in short, proposed in [47].…”

“…Finally, we give explicit computations for mixed spatio-temporal Ornstein-Uhlenbeck processes [47], also called MSTOU processes, and Lévy-driven CARMA fields [22,51]. In particular, our calculations in the case of the MSTOU processes show that it is possible to determine the asymptotic normality of the generalized method of moments estimator, GMM in short, proposed in [47].…”

“…We refer the reader to [6] for a comprehensive introduction to ambit fields which provide a rich class of spatial-temporal models on R × R m . Overall, MMAF and ambit fields are used in many applications throughout different disciplines, like geophysics [38], brain imaging [40], physics [11], biology [8,10], economics and finance [3,5,22,46,47].…”

The Multifaceted Behavior of Integrated supOU Processes: The Infinite Variance Case

“…Barndorff-Nielsen & Stelzer (2013)). Moreover, the limiting behavior is important for statistical estimation (see Stelzer et al (2015), Nguyen & Veraart (2018)).…”