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Theory of Probability and Mathematical Statistics



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

M. K. Ilienko

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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.

Keywords: Multidimensional regression sequences, m-regression sequences of random variables, almost sure convergence of random series

Bibliography:
1. V. V. Buldygin and S. A. Solntsev, Functional Methods in Problems of the Summation of Random Variables, Kiev, Naukova Dumka'', 1989. (Russian)
2. V. V. Buldygin and S. A. Solntsev, Asymptotic Behavior of Linearly Transformed Sums of Random Variables, Kluwer Academic Publishers, Dordrecht, 1997.
3. 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.
4. 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.
5. 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.
6. V. V. Buldygin and M. K. Runovska, Sums Whose Terms Are Elements of Linear Random Regression Sequences, Lambert Academic Publishing, 2014.
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8. 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.