Multiparameter and set-indexed stochastic processes

由多维时间参数或更一般地由一类集合索引的随机过程可以用于模拟许多随机现象，并且近年来一直是许多研究活动的焦点。然而，从一个完全有序的索引集（时间）到一个部分有序的索引集（多维时间或一类集合）的过程引入了许多挑战。特别是，描述过程的动态特性（即过程如何演变）变得更加复杂。诸如鞅或更新性质等依赖于“过去”和“未来”概念的概念必须扩展到这个更一般的框架。这是特别重要的，因为鞅方法提供优雅和强大的非参数方法的点过程推断，生存分析，状态估计，变点问题，并很容易将删失数据。 在本研究中，我们将进一步发展多参数鞅的基本性质，然后我们将利用这一理论开发新的统计技术，用于分析由多维时间参数索引的随机过程的各种例子。这类过程包括多参数时间序列、更新过程和经验过程。此外，多参数鞅方法可以应用于已被随机集删失的数据的生存分析（即该过程在随机集之外被掩盖）。我们还将研究匡威问题，其中随机过程是完全观察到的，但其行为可以在无法观察到的随机集上发生变化。在这种情况下，我们将开发技术来检测所谓的变更集的存在。 这些技术将在经济、金融、地理、地质、生物和环境等不同领域得到应用。

Random processes indexed by a multidimensional time parameter, or more generally by a class of sets, may be used to model many stochastic phenomena and have been the focus of much research activity in recent years. However, the passage from a totally ordered index set (time) to one that is partially ordered (multidimensional time or a class of sets) introduces many challenges. In particular, describing the dynamical properties of the process (i.e. how the process evolves) becomes far more complex. Concepts such as the martingale or renewal property that depend on the ideas of "past" and "future" must be extended to this more general framework. This is particularly important since martingale methods provide elegant and powerful nonparametric methods for point process inference, survival analysis, state estimation, change point problems, and easily incorporate censored data. In this research, we will be further developing fundamental properties of multi-parameter martingales, and then we will exploit this theory to develop new statistical techniques for the analysis of various examples of stochastic processes indexed by a multidimensional time parameter. Such processes include multi-parameter time series, renewal processes and empirical processes. Furthermore, multi-parameter martingale methods can be applied to survival analysis of data that has been censored by a random set (i.e. the process is obscured outside of a random set). We will also examine the converse problem, in which the stochastic process is totally observed, but its behaviour can change on a random set that cannot be observed. In this case, we will develop techniques to detect the existence of the so-called change set. These techniques will have applications in diverse areas, including economics, finance, geography, geology, biology and the environment.