Abstract. We study models of finite mixed empirical ordered point processes in compact metric spaces constructed from samples without repetition. We introduce the notion of the generating sequence of the probability measure of an ordered point process. A multidimensional family of distributions is constructed that completely determines the probability distribution of an ordered point process. An example is considered where we evaluate multidimensional distributions.Interest in the theory of stochastic point processes and marked point processes has been continually growing over the last two decades in view of their applications in stochastic geometry, economics, biology, ecology, cosmology, and stereology. However the models of marked point processes with statistical interactions between marked pairs, points of positions, and marks have not yet been studied in full detail. When simulating on computers some problems of spherical stochastic geometry, one faces the problem of constructing the trajectories of various types of point processes and marked point processes [6,12].In this paper, we consider models of point processes and marked point processes in compact metric spaces constructed from random samples without repetition. The models of this kind are called mixed empirical ordered point processes and marked point processes [10].The main definitions of the theory of finite simple ordered point processes and marked point processes are given in Section 1. The structure of finite simple point processes and marked point processes in compact metric spaces is studied in Section 2. We introduce the generating sequence of the probability measure of a point process (marked point process). In Section 3, we construct a model of a finite simple mixed empirical ordered point process in a compact metric space whose trajectories are samples without repetition from a population X according to the distribution P x on the σ-algebra A X .We construct a multidimensional family of distributions that completely determines the probability distribution P of a point process. We also consider an example and evaluate multidimensional distributions.

Abstract. Models of finite simple mixed empirical ordered marked point processes in compact metric spaces are studied in the paper. The processes are constructed from simple samples drawn without replacement from a population. The notion of an ordered marked point process with independent and 1-dependent marks is introduced. Examples of ordered marked point processes with independent and 1-dependent marks are given.

Moment measures of mixed empirical random point processes and marked point processes are investigated by using probability generating functions of random counting measures. 2010 Mathematics Subject Classification. Primary 60G55. Key words and phrases. Mixed empirical point processes, marked point processes, probability generating functions, moment measures.