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Theory of Probability and Mathematical Statistics

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A refinement of conditions for the almost sure convergence of series of multidimensional regression sequences


Author: M. K. Ilienko
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 93 (2015).
Journal: Theor. Probability and Math. Statist. 93 (2016), 71-78
MSC (2010): Primary 60G50, 65B10, 60G15; Secondary 40A05
DOI: https://doi.org/10.1090/tpms/994
Published electronically: February 7, 2017
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Abstract: We obtain a general criterion for the almost sure convergence of a series whose terms are elements of a multidimensional autoregressive sequence with arbitrary matrix coefficients. In particular, the case of degenerate matrices is also considered. This result extends an earlier result by Buldygin and Runovska who obtained necessary and sufficient conditions for the almost sure convergence of a random series whose terms are elements of a multidimensional Gaussian Markov sequence with nondegenerate matrix coefficients.


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Additional Information

M. K. Ilienko
Affiliation: Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine “KPI”, Peremogy Avenue, 37, Kyiv 03056, Ukraine
Email: matan@kpi.ua

DOI: https://doi.org/10.1090/tpms/994
Keywords: Multidimensional regression sequences, $m$-regression sequences of random variables, almost sure convergence of random series
Received by editor(s): June 14, 2015
Published electronically: February 7, 2017
Additional Notes: Supported by Swiss National Science Foundation, grant N IZ73Z0_152292
The paper was prepared following the talk at the International Conference “Probability, Reliability and Stochastic Optimization (PRESTO-2015)” held in Kyiv, Ukraine, April 7–10, 2015
Article copyright: © Copyright 2017 American Mathematical Society

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