A refinement of conditions for the almost sure convergence of series of multidimensional regression sequences
Author:
M. K. Ilienko
Translated by:
N. Semenov
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
MathSciNet review:
3553440
Full-text PDF Free Access
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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.
References
- V. V. Buldygin and S. A. Solntsev, Funktsional′nye metody v zadachakh summirovaniya sluchaĭ nykh velichin, “Naukova Dumka”, Kiev, 1989 (Russian). MR 1007589
- 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, DOI 10.1007/978-94-011-5568-7
- 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
- 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
- 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, DOI 10.1080/03610926.2011.581163
- V. V. Buldygin and M. K. Runovska, Sums Whose Terms Are Elements of Linear Random Regression Sequences, Lambert Academic Publishing, 2014.
- 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, DOI 10.1090/S0094-9000-2012-00848-X
- 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, DOI 10.1090/S0094-9000-2012-00857-0
References
- V. V. Buldygin and S. A. Solntsev, Functional Methods in Problems of the Summation of Random Variables, Kiev, “Naukova Dumka”, 1989. (Russian) MR 1007589
- V. V. Buldygin and S. A. Solntsev, Asymptotic Behavior of Linearly Transformed Sums of Random Variables, Kluwer Academic Publishers, Dordrecht, 1997. MR 1471203
- V. V. Buldygin and M. K. Runovska, On the convergence of series of autoregressive sequences, Theory Stoch. Process. 15(31) (2010), no. 1, pp. 7–14. MR 2603166
- V. V. Buldygin and M. K. Runovska, On the convergence of series of autoregressive sequences in Banach spaces, Theory Stoch. Process. 16(32) (2010), no. 1, pp. 29–38. MR 2779843
- V. V. Buldygin and M. K. Runovska, Almost sure convergence of the series of Gaussian Markov sequences, Comm. Statist. Theory Methods 40 (2011), no. 19–20, pp. 3407–3424. MR 2860747
- V. V. Buldygin and M. K. Runovska, Sums Whose Terms Are Elements of Linear Random Regression Sequences, Lambert Academic Publishing, 2014.
- M. K. Runovs’ka, Convergence of series composed of elements of Gaussian Markov sequences, Teor. Ĭmovir. Mat. Stat. 83 (2010), 125–137; English transl. in Theory Probab. Math. Statist. 83 (2011), 149–162. MR 2768855
- M. K. Runovs’ka, Convergence of series of elements of multidimensional Gaussian Markov sequences, Teor. Ĭmovir. Mat. Stat. 84 (2011), 131–141; English transl. in Theory Probab. Math. Statist. 84 (2012), 139–150. MR 2857424
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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
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
