Characterizing Stationarities Using Distributions of Random Locations

平稳性是随机过程或随机场的分布的移位不变性。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广泛研究。虽然关于平稳性已经有很多了解，并且可以找到丰富的文献流，但是仍然存在与平稳性的关系可能是内在的但不容易察觉的性质。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的分布之间的关系，例如路径上确界/下确界的位置，命中时间，我们引入了一类随机位置，并证明了在平稳性下，这类随机位置的分布必须满足一组非常特殊的条件。进一步证明了这组条件实际上等价于过程的平稳性，在这个意义上，一个过程是平稳的，当且仅当该条件对于该族中的所有随机位置都满足。用随机位置的分布刻画了一维严格平稳性。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也存在类似的结果和特征。所提出的研究来自于观察到随机过程空间和随机位置集合之间存在一般的对偶性，在这个意义上，过程属于呈现某种类型平稳性的空间，当且仅当我们推导出的条件对相应随机位置集合中的所有成员成立。以往的工作表明，对于一维严格平稳过程和一维平稳增量过程，都可以建立这样的对偶。我建议将这些结果扩展到平稳性的其他概念，从而建立一个系统的方法来表征，甚至定义，通过确定不同类别的随机位置，不同的平稳性。第一示例可以包括相对平稳过程（其分布是移位不变的仅限于紧凑区间的过程）和弱意义平稳过程（其前两个矩是移位不变的过程）。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具有重要价值的应用。我还建议进一步发展随机位置理论，以及利用随机位置分布的性质构造各种平稳性的新的统计检验，所提出的工作将在概率论和统计学方面做出重大贡献。它导致了一个新的家庭的统计测试平稳性，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。如果没有本研究所取得的结果，这种试验就不能成立。随机位置的分布的性质也将导致随机位置的函数的期望的最优界，这可以用于各种优化问题。此外，随机位置的设置与有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的一些悬而未决的问题。最后但并非最不重要的是，随机位置的分布知识将有助于在应用领域，如信用风险管理和地震预测，因此，具有潜在的价值，以改善管理的几个行业，这是至关重要的加拿大公众。

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.