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

## TL;DR

This paper introduces a flexible spatio-temporal Ornstein-Uhlenbeck process capable of modeling both short-range and long-range dependence, with theoretical analysis, simulation algorithms, and inference methods for practical applications.

## Contribution

It extends the Lévy-driven Ornstein-Uhlenbeck process by varying the rate parameter, enabling modeling of mixed dependence structures and non-separable correlations.

## Key findings

- Theoretical properties like stationarity and moments are established.
- A simulation algorithm for the compound Poisson case is developed.
- Inference via the generalized method of moments is demonstrated.

## 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 L\'evy-driven spatio-temporal Ornstein-Uhlenbeck process by randomly varying its rate parameter to model both short-range and long-range 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 L\'evy bases. The generalised method of moments is used for inference and simulation experiments are conducted with a view towards asymptotic properties.