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

Runovska, M. K.

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.

Reviewer: Aleksander Iksanov (Kiev)

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

Keywords:

Gaussian Markov sequence of random variables; almost sure convergence; random series; theory of sums of independent random elements with operator normalizations

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].