Point processes and, more generally, random measures are ubiquitous in modern statistics. However, they can only take positive values, which is a severe limitation in many situations. In this work, we introduce and study random signed measures, also known as real-valued random measures, and apply them to constrcut various Bayesian non-parametric models. In particular, we provide an existence result for random signed measures, allowing us to obtain a canonical definition for them and solve a 70-year-old open problem. Further, we provide a representation of completely random signed measures (CRSMs), which extends the celebrated Kingman's representation for completely random measures (CRMs) to the real-valued case. We then introduce specific classes of random signed measures, including the Skellam point process, which plays the role of the Poisson point process in the real-valued case, and the Gaussian random measure. We use the theoretical results to develop two Bayesian nonparametric models -- one for topic modeling and the other for random graphs -- and to investigate mean function estimation in Bayesian nonparametric regression.

翻译：点过程以及更一般的随机测度在现代统计学中无处不在。然而，它们只能取正值，这在许多情况下是一个严重的限制。在这项工作中，我们引入并研究了随机有符号测度，也称为实值随机测度，并将其应用于构建各种贝叶斯非参数模型。特别地，我们给出了随机有符号测度的一个存在性结果，这使我们能够为其获得一个规范定义，并解决了一个存在70年的开放性问题。此外，我们给出了完全随机有符号测度的一种表示，这将在实值情形下将著名的Kingman完全随机测度表示推广到实值情形。接着，我们引入了随机有符号测度的具体类别，包括在实值情形下扮演泊松点过程角色的Skellam点过程，以及高斯随机测度。我们利用这些理论结果开发了两个贝叶斯非参数模型——一个用于主题建模，另一个用于随机图——并研究了贝叶斯非参数回归中的均值函数估计。