Stationarity And Random Locations
We introduce a family of random locations called "intrinsic location functionals", which include most of the random locations that one may encounter in many cases, e.g., the location of the path supremum/infimum over an interval, the first/last hitting times, etc. It is proved that the distributions of these locations must satisfy certain properties, such as the absolute continuity in the interior of the interval, and a group of constraints on the total variation of the density function. It is further shown that the list of properties that we obtained for the distributions is actually equivalent to the stationarity of the process, in the sense that a process is stationary if and only if the distributions of all intrinsic location functionals satisfy the list of properties. In this way we get an alternative characterization of stationarity from the perspective of random locations. Moreover, we develop alternative equivalent descriptions for intrinsic location functionals in terms of partially ordered random point sets and piecewise linear functions. The main results can be extended in many directions, for instance, stationary increment processes, stationary random fields and isotropic random fields.
stationary process; random location; total variation
Nussbaum, Michael; Jarrow, Robert A.
Ph.D. of Operations Research
Doctor of Philosophy
dissertation or thesis