Characterizing Stationarities Using Distributions of Random Locations

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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.