Mass-stationarity through the Cox process
Seminar Room 1, Newton Institute
AbstractConsider a random measure on a locally compact Abelian group, for instance the d-dimensional Euclidean space. Consider also a random element in a measurable space on which the group acts, for instance a random field indexed by the group. Mass-stationarity of the random element with respect to the measure is an intrinsic characterization of Palm versions with respect to stationary random measures. It is a formalization of the intuitive idea that the origin is a typical location in the mass of the measure. Mass-stationarity is an extension to random measures of point stationarity with respect to a simple point process.
A Cox process represents the mass of a random measure through a collection of points placed independently at typical locations in the mass. Thus if the random measure is mass-stationary and we add an extra point at the origin to the Cox process then the points of that modified Cox process are all at typical locations in the mass of the random measure. It turns out that mass-stationarity with respect to the random measure reduces to mass-stationarity with respect to the modified Cox process. In particular, for diffuse random measures mass-stationarity reduces in this way to point stationarity.