It is known in the Spatial Statistics literature that the choice of the kernel function determines the spatial covariance function of moving average or process convolution models. In the previous two projects, we explored how the shape of the ambit sets and the introduction of a stochastic volatility term affects the properties of the ambit field. Now, we return to the set up of the spatio-temporal Ornstein-Uhlenbeck (STOU) process and generalise the model by allowing the rate parameter (i.e. the decay parameter in the exponential kernel) to vary randomly. This is achieved by introducing a probability density function to the *control measure of the Lévy basis*.

This extended model, i.e the mixed STOU (MSTOU) process, provides a bridge between the short-range and *long-range dependence*. While the former is simpler to assume in practice, the latter has been observed in data from finance, hydrology, geophysics and economics.

A paper based on this project with Almut has been published at Stochastics and the corresponding R code is available online. There, we establish theoretical properties of MSTOU processes and develop a simulation algorithm for the compound Poisson case which can be used to approximate other Lévy bases. We also conduct simulation experiments involving the generalised method of moments so as to shine light on asymptotic properties.