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Introduction to Random Fields

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Stochastic Geometry, Spatial Statistics and Random Fields

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2068))

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.

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Notes

  1. 1.

    The notation T comes from “time”, since for random processes t ∈ T is often interpreted as the time parameter.

  2. 2.

    The important contributions of other researchers to establishing this result are discussed in [520, pp. 69–70].

  3. 3.

    Clearly Z is finite if S is a finite set.

  4. 4.

    Note that the canonical potential depends on o ∈ S.

  5. 5.

    Because in ordinary language a clique is a group of people who know and favour each other.

  6. 6.

    All densities are considered with respect to the corresponding Lebesgue measures.

  7. 7.

    For the sake of simplicity, we do not consider the case when T is a subset of a linear vector space.

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  1. Moscow State University, Moscow, Russia

    Alexander Bulinski

  2. Ulm University, Ulm, Germany

    Evgeny Spodarev

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    Evgeny Spodarev

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Bulinski, A., Spodarev, E. (2013). Introduction to Random Fields. In: Spodarev, E. (eds) Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Mathematics, vol 2068. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33305-7_9

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