A sufficient condition for linear growth of variances in a stationary random sequence
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- by Richard C. Bradley PDF
- Proc. Amer. Math. Soc. 83 (1981), 586-589 Request permission
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 $\tau (n) = \operatorname {Sup} \{ |\operatorname {Corr} (\sum _{k = - I}^0{X_k},\sum _{k = m}^{m + I}{X_k})|:m \geqslant n,I \geqslant 0\}$. 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
- István Berkes and Walter Philipp, Approximation theorems for independent and weakly dependent random vectors, Ann. Probab. 7 (1979), no. 1, 29–54. MR 515811
- I. A. Ibragimov, A remark on the central limit theorem for dependent random variables, Teor. Verojatnost. i Primenen. 20 (1975), 134–140 (Russian, with English summary). MR 0362448
- Il′dar Abdullovich Ibragimov and Y. A. Rozanov, Gaussian random processes, Applications of Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Translated from the Russian by A. B. Aries. MR 543837 —, Gaussian random processes, "Nauka", Moscow, 1970. (Russian)
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 83 (1981), 586-589
- MSC: Primary 60G10; Secondary 60F05, 62M10
- DOI: https://doi.org/10.1090/S0002-9939-1981-0627698-5
- MathSciNet review: 627698