## Convergence of series of Gaussian Markov sequences.(English. Russian original)Zbl 1253.65003

Theory Probab. Math. Stat. 83, 149-162 (2011); translation from Teor. Jmovirn. Mat. Stat. 83, 125-137 (2010).
Let $$(\xi_k)_{k\in\mathbb{N}_0}$$ be a random sequence defined by $\xi_0=0, \;\;\xi_k=\alpha_k\xi_{k-1}+\beta_k\gamma_k, \;\;k\in\mathbb{N},$ where $$(\alpha_k)_{k\in \mathbb{N}}$$ and $$(\beta_k)_{k\in \mathbb{N}}$$ are real and nonnegative numbers, respectively, and $$(\gamma_k)_{k\in \mathbb{N}}$$ are i.i.d. random variables with standard normal distribution. The author proves a criterion for the a.s. convergence of the random series $$\sum_{k\geq 1}\xi_k$$. The main technical tool is a reduction to a better known two-dimensional stochastic difference equation $$X_k=A_kX_{k-1}+B_k$$. Several illustrating examples are also given.

### MSC:

 65B10 Numerical summation of series 60F15 Strong limit theorems 60G50 Sums of independent random variables; random walks 60G15 Gaussian processes 40A05 Convergence and divergence of series and sequences
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### References:

 [1] V. V. Buldygin, The strong law of large numbers and the convergence to zero of Gaussian sequences, Teor. Verojatnost. i Mat. Statist. 19 (1978), 33 – 41, 156 – 157 (Russian, with English summary). [2] Функционал$$^{\приме}$$ные методы в задачах суммирования случайных величин, ”Наукова Думка”, Киев, 1989 (Руссиан). [3] 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. · Zbl 0906.60002 [4] Valerii V. Buldygin and Marina K. Runovska, On the convergence of series of autoregressive sequences, Theory Stoch. Process. 15 (2009), no. 1, 7 – 14. · Zbl 1224.60093 [5] Теория рядов, ”Наука”, Мосцощ, 1979 (Руссиан). Фоуртх едитион, ревисед анд аугментед; Избранные главы высшей математики для инженеров и студентов втузов. [Селецтед Чаптерс ин Хигхер Матхематицс фор Енгинеерс анд Течницал Университы Студенц].
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