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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**V. V. Buldygin and S. A. Solntsev,*\cyr Funktsional′nye metody v zadachakh summirovaniya sluchaĭnykh velichin*, “Naukova Dumka”, Kiev, 1989 (Russian). MR**1007589****2.**Valery Buldygin and Serguei Solntsev,*Asymptotic behaviour of linearly transformed sums of random variables*, Mathematics and its Applications, vol. 416, Kluwer Academic Publishers Group, Dordrecht, 1997. Translated from the 1989 Russian original by Vladimir Zaiats and revised, updated and expanded by the authors. MR**1471203****3.**Valerii V. Buldygin and Marina K. Runovska,*On the convergence of series of autoregressive sequences*, Theory Stoch. Process.**15**(2009), no. 1, 7–14. MR**2603166****4.**Valerii V. Buldygin and Marina K. Runovska,*On the convergence of series of autoregressive sequences in Banach spaces*, Theory Stoch. Process.**16**(2010), no. 1, 29–38. MR**2779843****5.**Valerii V. Buldygin and Marina K. Runovska,*Almost sure convergence of the series of Gaussian Markov sequences*, Comm. Statist. Theory Methods**40**(2011), no. 19-20, 3407–3424. MR**2860747**, https://doi.org/10.1080/03610926.2011.581163**6.**V. V. Buldygin and M. K. Runovska,*Sums Whose Terms Are Elements of Linear Random Regression Sequences*, Lambert Academic Publishing, 2014.**7.**M. K. Runovs′ka,*Convergence of series composed of elements of Gaussian Markov sequences*, Teor. Ĭmovīr. Mat. Stat.**83**(2010), 125–137 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist.**83**(2011), 149–162. MR**2768855**, https://doi.org/10.1090/S0094-9000-2012-00848-X**8.**M. K. Runovs′ka,*Convergence of series of elements of multidimensional Gaussian Markov sequences*, Teor. Ĭmovīr. Mat. Stat.**84**(2011), 131–141 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist.**84**(2012), 139–150. MR**2857424**, https://doi.org/10.1090/S0094-9000-2012-00857-0

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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