Stationary Marked Point Processes

Abstract

Many areas of engineering and statistics involve the study of a sequence of random events, described by points occurring over time (or space), together with a mark for each such point that contains some further information about it (type, class, etc.). Examples include image analysis, stochastic geometry, telecommunications, credit or insurance risk, discrete-event simulation, empirical processes, and general queueing theory. In telecommunications, for example, the events might be the arrival times of requests for bandwidth usage, and the marks the bandwidth capacity requested. In a mobile phone context, the points could represent the locations (at some given time) of all mobile phones, and the marks 1 or 0 as to whether the phone is in use or not. Such a stochastic sequence is called a random marked point process, an MPP for short. In a stationary stochastic setting (e.g., if we have moved our origin far away in time or space, so that moving further would not change the distribution of what we see) there are two versions of an MPP of interest depending on how we choose our origin: point-stationary and time-stationary (space-stationary). The first randomly chooses an event point as the origin, whereas the second randomly chooses a time (or space) point as the origin. Fundamental mathematical relationships exists between these two versions allowing for nice applications and computations. In what follows, we present this basic theory with emphasis on one-dimensional processes over time, but also include some recent results for d-dimensional Euclidean space, .

This chapter will primarily deal with marked point processes with points on the real line (time). Spatial point processes with points in will be touched upon in the final section; some of the deepest results in multiple dimensions have only come about recently.

Topics covered include point- and time-stationarity, inversion formulas, the Palm distribution, Campbellʼs formula, MPPs jointly with a stochastic process, the rate conservation law, conditional intensities, and ergodicity.