Intrinsically Weighted Means of Marked Point Processes

Abstract: For a non-stationary or non-ergodic marked point process (MPP) on $\R^d$, the definition of averages becomes ambiguous as the process might have a different stochastic behavior in different realizations (non-ergodicity) or in different areas of the observation window (non-stationarity). We investigate different definitions for the moments, including a new hierarchical definition for non-ergodic MPPs, and embed them into a family of weighted mean marks. We point out examples of application in which different weighted mean marks all have a sensible meaning. Further, asymptotic properties of the corresponding estimators are investigated as well as optimal weighting procedures.