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A sufficient condition for linear growth of variances in a stationary random sequence

Author: Richard C. Bradley
Journal: Proc. Amer. Math. Soc. 83 (1981), 586-589
MSC: Primary 60G10; Secondary 60F05, 62M10
MathSciNet review: 627698
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Abstract: Suppose $ ({X_k},k = \ldots , - 1,0,1, \ldots )$ is a weakly stationary random sequence. For each positive integer $ n{\text{let }}{S_n} \equiv {X_1} + \cdots + {X_n}$ and $ \tau (n) = \operatorname{Sup} \{ \vert\operatorname{Corr} (\sum _{k = - I}^0{X_k},\sum _{k = m}^{m + I}{X_k})\vert:m \geqslant n,I \geqslant 0\} $. If Var $ {S_n} \to \infty $ as $ n \to \infty $ and $ \sum _{n = 0}^\infty \tau ({2^n}) < \infty $, then $ {n^{ - 1}}$ Var $ {S_n}$ converges to a finite positive limit as $ n \to \infty $. A bound on the rate of convergence is obtained.

References [Enhancements On Off] (What's this?)

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Keywords: Weakly stationary, variance of partial sums, spectral density
Article copyright: © Copyright 1981 American Mathematical Society

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