  This paper proposes a novel family of geostatistical models to account for
features that cannot be properly accommodated by traditional Gaussian
processes. The family is specified hierarchically and combines the
infinite-dimensional dynamics of Gaussian processes with that of any
multivariate continuous distribution. This combination is stochastically
defined through a latent Poisson process and the new family is called the
Poisson-Gaussian Mixture Process - POGAMP. Whilst the attempt of defining
geostatistical processes by assigning some arbitrary continuous distribution to
be the finite-dimension distributions usually leads to non-valid processes, the
finite-dimensional distributions of the POGAMP can be arbitrarily close to any
continuous distribution and still define a valid process. Formal results to
establish the existence and some important properties of the POGAMP, such as
absolute continuity with respect to a Gaussian process measure, are provided.
Also, an MCMC algorithm is carefully devised to perform Bayesian inference when
the POGAMP is discretely observed in some space domain.
