Characterizing Stationarities Using Distributions of Random Locations

平稳性是随机过程或随机场分布的移位不变性。它在概率论、统计学及其应用中起着至关重要的作用，长期以来一直受到人们的深入研究。尽管我们对平稳性有很多了解，并且可以找到丰富的文献流，但仍然存在与平稳性相关的属性，这些属性可能是内在的，但不容易被感知。最近，申请人与Gennady Samorodnitsky合作，研究了随机过程的严格平稳性与随机位置分布之间的关系，例如路径的最优/最小值的位置，命中次数等。我们引入了一组随机位置，并证明了在平稳性条件下，这组随机位置的分布必须满足一组非常特定的条件。进一步证明了这组条件实际上等价于过程的平稳性，即当且仅当一族中所有随机位置的条件都满足时，过程是平稳的。这样，我们用随机位置的分布来表征一维严格平稳性。后来，我们还发现在过程增量的严格平稳性和这些随机位置的子类之间存在类似的结果和表征。我们提出的研究来自于对随机过程空间和随机位置集合之间存在一般对偶的观察，也就是说，当且仅当我们推导的条件对相应的随机位置集合中的所有成员都成立时，一个过程属于具有某种平稳性的空间。以往的研究表明，对于一维严格平稳过程和一维平稳增量过程，可以建立这样的对偶性。我建议将这些结果扩展到平稳性的其他概念，从而通过识别不同类别的随机位置来建立一种系统的方法来表征甚至定义不同的平稳性。第一个例子可以包括相对平稳过程（其分布仅局限于紧区间的移位不变量的过程）和弱感知平稳过程（其前两个矩是移位不变量的过程）。这两种方法都被广泛用于模拟信号、水位、经济数据和许多其他对加拿大有重要价值的应用。我还建议进一步发展随机位置理论，并利用随机位置分布的性质构造各种平稳性的新统计检验。拟议的工作将在概率和统计两方面作出重大贡献。它导致了一系列新的统计检验的平稳性，这将是特别有用的，当只有部分信息，如某些事件的发生时间，是已知的过程。如果没有本研究获得的结果，就不能建立这样的测试。随机位置分布的性质也会导致随机位置函数期望的最优界，这可以用于各种优化问题。此外，随机地点的设置与有最后期限的排队系统之间存在着天然的联系，本研究的结果可以回答排队理论中某些悬而未决的问题。最后但并非最不重要的是，随机地点的分布知识将有助于信用风险管理和地震预测等应用领域，因此对改善对加拿大公众至关重要的几个行业的管理具有潜在价值。

Stationarity is the shift invariance of the distribution of a stochastic process or a random field. It plays an essential role in probability, statistics and their applications, and has been intensively studied for a long time. Although much is known about stationarity and a rich stream of literature can be found, there remain properties whose relation to stationarity may be intrinsic yet not easy to perceive.Recently, the applicant, collaborating with Gennady Samorodnitsky, has investigated the relation between strict stationarity of stochastic processes and the distributions of random locations, such as the location of the path supremum/infimum, hitting times, etc. We introduced a family of random locations and proved that under stationarity, the distribution of any random location in this family must satisfy a group of very specific conditions. It was further proved that this group of conditions is actually equivalent to the stationarity of the process, in the sense that a process is stationary if and only if the conditions are satisfied for all the random locations in the family. In this way we characterized the one dimensional strict stationarity by the distributions of random locations. Later, we also discovered that similar results and characterization exist between the strict stationarity of the increments of a process and a subclass of these random locations.The proposed research comes from the observation of the existence of a general duality between the spaces of stochastic processes and the sets of random locations, in the sense that a process belongs to some space presenting a certain type of stationarity, if and only if the conditions that we derived hold for all the members in a corresponding set of random locations. The previous works show that such a duality can be established for one dimensional strict stationary processes and one dimensional stationary increment processes. I propose to extend these results to other notions of stationarity, thus building a systemic way to characterize, and even define, different stationarities by identifying different classes of random locations. First examples may include relatively stationary processes (processes whose distributions are shift invariant only restricted to a compact interval) and weak-sense stationary processes (processes whose first two moments are shift invariant). Both of these are widely used to model signals, water levels, economic data and many other applications of significant value to Canada. I also propose to make a further development of the theory of random locations, as well as to construct new statistical tests for various stationarities using the properties of the distributions of random locations.The proposed work will contribute significantly in both probability and statistics. It leads to a new family of statistical tests for stationarities, which will be especially useful when only partial information, such as the occurrence time of certain event, is known for a process. Such tests can not be established without the results obtained in this research. The properties of the distributions of random locations will also result in optimal bounds for the expectation of the functions of random locations, which can be used in various optimization problems. Moreover, there is a natural link between the setting of random locations and queueing systems with deadlines, and the results of this research can answer certain pending questions in queueing theory. Last but not least, the distributional knowledge of random locations will be helpful in applied areas such as credit risk management and earthquake prediction, therefore has potential value for improving the management of several industries which are critical to the Canadian public.