Introduction to Random Fields
This chapter gives preliminaries on random fields necessary for understanding of the next two chapters on limit theorems. Basic classes of random fields (Gaussian, stable, infinitely divisible, Markov and Gibbs fields, etc.) are considered. Correlation th
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Introduction to Random Fields Alexander Bulinski and Evgeny Spodarev
Abstract This chapter gives preliminaries on random fields necessary for understanding of the next two chapters on limit theorems. Basic classes of random fields (Gaussian, stable, infinitely divisible, Markov and Gibbs fields, etc.) are considered. Correlation theory of stationary random functions as well as elementary nonparametric statistics and an overview of simulation techniques are discussed in more detail.
9.1 Random Functions Let .˝; A; P/ be a probability space and .S; B/ be a measurable space (i.e. we consider an arbitrary set S endowed with a sigma-algebra B). We always assume that ˝ ¤ ; and S ¤ ;. Definition 9.1. A random element W ˝ ! S is an AjB-measurable mapping (one writes 2 AjB), that is, 1 .B/ WD f! 2 ˝ W .!/ 2 Bg 2 A for all B 2 B:
(9.1)
If is a random element, then for a given ! 2 ˝, the value .!/ is called a realization of . We say that the sigma-algebra B consisting of some subsets of S is generated by a system M of subsets of S if B is the intersection of all -algebras (of subsets of S )
A. Bulinski () Moscow State University, Moscow, Russia e-mail: [email protected] E. Spodarev Ulm University, Ulm, Germany e-mail: [email protected] E. Spodarev (ed.), Stochastic Geometry, Spatial Statistics and Random Fields, Lecture Notes in Mathematics 2068, DOI 10.1007/978-3-642-33305-7 9, © Springer-Verlag Berlin Heidelberg 2013
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containing M [ fS g. One uses the notation B D fMg. For topological (or metric) space S one usually takes B D B.S / where B.S / is the Borel -algebra. Recall that B.S / is generated by all open subsets of S . If S D Rn and B D B.Rn/ where n 2 N, then random element is called a random variable when n D 1 and a random vector when n > 1. One also says that is a random variable with values in a state-space S (endowed with -algebra B) if (9.1) holds. Exercise 9.1. Let .˝; A/ and .S; B/ be measurable spaces and B D fMg with M being a family of subsets of S . Prove that a mapping W ˝ ! S is AjBmeasurable iff 1 .C / 2 A for any C 2 M. Example 9.1 (Point process). Let N be the set of all locally finite simple point d patterns ' D fxi g1 i D1 R , cf. Sect. 3.1.1. It means '.B/ WD j' \ Bj < 1 for d any bounded set B 2 B.R / (one writes B 2 B0 .Rd /) where jAj stands for the cardinality of A and we assume that xi ¤ xj for i ¤ j . Let N be the minimal -algebra generated in N by all sets of the form f' 2 N W '.B/ D kg for k 2 ZC and B 2 B0 .Rd /. Take .S; B/ D .N ; N/. The point process W ˝ ! N is an AjN-measurable random element. Another possibility to define is to use a random counting measure .!; B/ D
1 X
ıxi .!/ .B/; ! 2 ˝; B 2 B0 .Rd /;
(9.2)
i D1
where ıx is the Dirac measure concentrated at a point x and a point process fxi .!/g can be viewed as a support of this measure, see Sect. 4.1.1 (Fig. 9.1). Example 9.2 (Random closed sets). Let F be the family of all closed sets in Rd . Introduce -algebra F generated by th
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