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Quenched Invariance Principles for Orthomartingale-Like Sequences

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Abstract

In this paper, we study the central limit theorem and its functional form for random fields which are started not from their equilibrium, but rather under the measure conditioned by the past sigma field. The initial class considered is that of orthomartingales and then the result is extended to a more general class of random fields by approximating them, in some sense, with an orthomartingale. We construct an example which shows that there are orthomartingales which satisfy the CLT but not its quenched form. This example also clarifies the optimality of the moment conditions used for the validity of our results. Finally, by using the so-called orthomartingale-coboundary decomposition, we apply our results to linear and nonlinear random fields.

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Acknowledgements

This research was supported in part by the NSF Grant DMS–1811373. The first author would like to offer special thanks to the University of Rouen Normandie, where a large portion of this research was accomplished during her visit of the Department of Mathematics. The authors wish to thank the referee for many in-depth comments, suggestions and corrections, which have greatly improved the manuscript.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, OH, 45221-0025, USA

    Magda Peligrad

  2. LMRS, CNRS and Université de Rouen Normandie, Saint-Étienne-du-Rouvray, France

    Dalibor Volný

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  1. Magda Peligrad
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  2. Dalibor Volný
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Correspondence to Magda Peligrad.

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Peligrad, M., Volný, D. Quenched Invariance Principles for Orthomartingale-Like Sequences. J Theor Probab 33, 1238–1265 (2020). https://doi.org/10.1007/s10959-019-00914-z

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  • DOI: https://doi.org/10.1007/s10959-019-00914-z

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