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

Published online by Cambridge University Press:  20 November 2018

Byron Schmuland  and
Wei Sun
Byron Schmuland
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1 e-mail: schmu@stat.ualberta.ca
Wei Sun
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1 e-mail: wsun@stat.ualberta.ca and Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100080, China
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Abstract

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In [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 $\phi $-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.

Keywords

60F9960G60law of the iterated logarithm
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 1 , 01 February 2004 , pp. 209 - 224
Copyright
Copyright © Canadian Mathematical Society 2004

References

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