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.

翻译：本文提出了一类新型地统计模型，以处理传统高斯过程无法妥善建模的特征。该模型族通过分层方式定义，结合了高斯过程的无穷维动力学与任意多元连续分布的分布特性。这种结合通过隐泊松过程随机定义，新模型族称为泊松-高斯混合过程（Poisson-Gaussian Mixture Process, POGAMP）。尽管将任意连续分布赋予有限维分布来定义地统计过程的尝试通常导致非有效过程，但POGAMP的有限维分布可以任意接近任何连续分布，同时仍能定义有效过程。我们提供了建立POGAMP存在性的正式结果及其重要性质（如相对于高斯过程测度的绝对连续性）。此外，我们精心设计了一种MCMC算法，用于在空间域离散观测POGAMP时进行贝叶斯推断。