scholarly journals A Dependent Lindeberg Central Limit Theorem for Cluster Functionals on Stationary Random Fields

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 212
Author(s):  
José G. Gómez-García ◽  
Christophe Chesneau
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Field ◽  
Central Limit ◽  
Random Fields ◽  
Empirical Processes ◽  
Index Set ◽  
Finite Dimensional ◽  
Lindeberg Condition ◽  
Stationary Random Field

In this paper, we provide a central limit theorem for the finite-dimensional marginal distributions of empirical processes (Zn(f))f∈F whose index set F is a family of cluster functionals valued on blocks of values of a stationary random field. The practicality and applicability of the result depend mainly on the usual Lindeberg condition and on a sequence Tn which summarizes the dependence between the blocks of the random field values. Finally, in application, we use the previous result in order to show the Gaussian asymptotic behavior of the proposed iso-extremogram estimator.

scholarly journals A Central Limit Theorem for Spatial Observations

2016 ◽  
Vol 41 (3) ◽  
Author(s):  
István Fazekas ◽  
Zsolt Karácsony ◽  
Renáta Vas
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Field ◽  
Central Limit ◽  
Random Fields ◽  
Irregular Domains ◽  
The Central Limit Theorem

The Central Limit Theorem is proved for m-dependent random fields. The random field is observed in a sequence of irregular domains. The sequence of domains is increasing and at the same time the locations of the observations become more and more dense in the domains.

scholarly journals The Transparent Dead Leaves Model

2012 ◽  
Vol 44 (01) ◽  
pp. 1-20 ◽  
Author(s):  
B. Galerne ◽  
Y. Gousseau
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Field ◽  
Central Limit ◽  
Gaussian Random Field ◽  
Natural Images ◽  
Simulation Algorithm ◽  
New Model ◽  
Grain Model ◽  
Dead Leaves Model

In this paper we introduce the transparent dead leaves (TDL) random field, a new germ-grain model in which the grains are combined according to a transparency principle. Informally, this model may be seen as the superposition of infinitely many semitransparent objects. It is therefore of interest in view of the modeling of natural images. Properties of this new model are established and a simulation algorithm is proposed. The main contribution of the paper is to establish a central limit theorem, showing that, when varying the transparency of the grain from opacity to total transparency, the TDL model ranges from the dead leaves model to a Gaussian random field.

A Central Limit Theorem and Law of the Iterated Logarithm for a Random Field with Exponential Decay of Correlations

2004 ◽  
Vol 56 (1) ◽  
pp. 209-224 ◽  
Author(s):  
Byron Schmuland ◽  
Wei Sun
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Field ◽  
Central Limit ◽  
Exponential Decay ◽  
Gibbs Measures ◽  
Strong Mixing ◽  
Iterated Logarithm ◽  
Exponential Decay Of Correlations ◽  
The Law

AbstractIn [6], Walter Philipp wrote that “… the law of the iterated logarithm holds for any process for which the Borel-Cantelli Lemma, the central limit theorem with a reasonably good remainder and a certain maximal inequality are valid.” Many authors [1], [2], [4], [5], [9] have followed this plan in proving the law of the iterated logarithm for sequences (or fields) of dependent random variables.We carry on this tradition by proving the law of the iterated logarithm for a random field whose correlations satisfy an exponential decay condition like the one obtained by Spohn [8] for certain Gibbs measures. These do not fall into the ϕ-mixing or strong mixing cases established in the literature, but are needed for our investigations [7] into diffusions on configuration space.The proofs are all obtained by patching together standard results from [5], [9] while keeping a careful eye on the correlations.

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