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Aims and ScopeOptimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics, and other sciences.The series Springer Optimization and Its Applications publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository work that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multiobjective programming, description of software packages, approximation techniques and heuristic approaches.

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.

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