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Central Limit Theorems for Weakly Dependent 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 is a primer on the limit theorems for dependent random fields. First, dependence concepts such as mixing, association and their generalizations are introduced. Then, moment inequalities for sums of dependent random variables are stated which yield e.g. the asymptotic behaviour of the variance of these sums which is essential for the proof of limit theorems. Finally, central limit theorems for dependent random fields are given. Applications to excursion sets of random fields and Newman’s conjecture in the absence of finite susceptibility are discussed as well.

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Notes

  1. 1.

    Such family { η t , t ∈ T } exists due to Theorem 9.1.

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  1. Moscow State University, Moscow, Russia

    Alexander Bulinski

  2. Ulm University, Ulm, Germany

    Evgeny Spodarev

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  1. Alexander Bulinski
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    Evgeny Spodarev

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Bulinski, A., Spodarev, E. (2013). Central Limit Theorems for Weakly Dependent 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_10

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