Convergence of series of elements of multidimensional Gaussian Markov sequences
Author:
M. K. Runovska
Translated by:
O. I. Klesov
Journal:
Theor. Probability and Math. Statist. 84 (2012), 139-150
MSC (2010):
Primary 60G50, 65B10, 60G15; Secondary 40A05
DOI:
https://doi.org/10.1090/S0094-9000-2012-00857-0
Published electronically:
August 2, 2012
MathSciNet review:
2857424
Full-text PDF Free Access
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Additional Information
Abstract: Necessary and sufficient conditions are found for the almost sure convergence of random series of elements of a multidimensional Gaussian Markov sequence.
References
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- V. V. Buldygin and S. A. Solntsev, A contraction principle and a strong law of large numbers for weighted sums, Teor. Veroyatnost. i Primenen. 31 (1986), no. 3, 516–529 (Russian). MR 866871
- V. V. Buldygin and S. A. Solntsev, The strong law of large numbers for sums of independent random vectors with operator normalizations and the convergence to zero of Gaussian sequences, Teor. Veroyatnost. i Primenen. 32 (1987), no. 2, 266–281 (Russian). MR 902755
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- V. V. Buldygin and S. A. Solntsev, Convergence of sums of independent random vectors with operator normalizations, Teor. Veroyatnost. i Primenen. 36 (1991), no. 2, 346–351; English transl., Theory Probab. Appl. 36 (1991), no. 2, 360–365 (1992). MR 1119509, DOI https://doi.org/10.1137/1136039
- 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
- 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
- V. A. Koval′, Asymptotic behavior of solutions of stochastic recurrence equations in ${\bf R}^d$, Ukrain. Mat. Zh. 43 (1991), no. 6, 829–833 (Russian, with Ukrainian summary); English transl., Ukrainian Math. J. 43 (1991), no. 6, 776–783 (1992). MR 1134409, DOI https://doi.org/10.1007/BF01058946
- 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 https://doi.org/10.1090/S0094-9000-2012-00848-X
References
- V. V. Buldygin, The strong law of large numbers and the convergence to zero of Gaussian sequences, Teor. Veroyatnost. i Mat. Statist. 19 (1978), 33–41; English transl. in Theor. Probability Math. Statist. 19 (1980), 35–43. MR 0494429 (58:13294)
- V. V. Buldygin and S. A. Solntsev, The contraction principle and the strong law of large numbers for weighted sums, Teor. Veroyatnost. Primenen. 31 (1986), no. 3, 516–529; English transl. in Theory Probab. Appl. 31 (1986), 452–462. MR 866871 (88c:60021)
- V. V. Buldygin and S. A. Solntsev, The strong law of large numbers for sums of independent random vectors with operator normalizations and null convergence of Gaussian sequences, Teor. Veroyatnost. Primenen. 32 (1987), no. 2, 266–281; English transl. in Theory Probab. Appl. 32 (1987), 243–256. MR 902755 (88i:60013)
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- V. V. Buldygin and S. A. Solntsev, On convergence of sums of operator-normed independent random vectors, Teor. Veroyatnost. Primenen. 36 (1991), no. 2, 346–351; English transl. in Theory Probab. Appl. 36 (1991), 360–365. MR 1119509 (92h:60042)
- V. V. Buldygin and S. A. Solntsev, Asymptotic Behavior of Linearly Transformed Sums of Random Variables, Kluwer Academic Publishers, Dordrecht, 1997. MR 1471203 (98m:60002)
- V. V. Buldygin and M. K. Runovska, On the convergence of series of autoregressive sequences, Theory Stoch. Process. 15(31) (2009), no. 1, 7–14. MR 2603166 (2011a:60116)
- 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, 29–38. MR 2779843 (2011k:60009)
- V. A. Koval’, Asymptotic behavior of solutions of stochastic recurrence equations in $\mathbb {R}^d$, Ukr. Matem. Zh. 43 (1991), no. 6, 829–833; English transl. in Ukrain. Math. J. 43 (1991), 776–779. MR 1134409 (92k:60148)
- M. K. Runovska, Convergence of series of Gaussian Markov sequences, Teor. Imovirnost. ta Matem. Statist. 83 (2010), 125–137; English transl. in Theor. Probability and Math. Statist. 83 (2011), 149–162. MR 2768855 (2011k:60165)
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Additional Information
M. K. Runovska
Affiliation:
Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine (“KPI”), Peremogy Avenue 37, Kyiv 03056, Ukraine
Email:
matan@kiev.ua
Keywords:
Multidimensional Gaussian Markov sequence,
Gaussian $m$-Markov sequence,
almost sure convergence of random series,
sums of independent random elements with operator normalizations
Received by editor(s):
November 30, 2010
Published electronically:
August 2, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
