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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Sums of stationary sequences cannot grow slower than linearly

Author: Harry Kesten
Journal: Proc. Amer. Math. Soc. 49 (1975), 205-211
MSC: Primary 60F15; Secondary 28A65
MathSciNet review: 0370713
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Abstract: It is shown that for a stationary sequence of random variables $ {X_1},{X_2}, \cdots $ one has

$\displaystyle \lim \inf {n^{ - 1}}\sum\limits_{i = 1}^n {{X_i} > 0} $

a.e. on the set $ \{ \Sigma _1^n{X_i} \to \infty ,n \to \infty \} $.

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

  • [1] L. Breiman, Probability, Addison-Wesley, Reading, Mass., 1968. MR 37 #4841. MR 0229267 (37:4841)
  • [2] D. Tanny, A zero-one law for stationary sequences, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 30 (1974), 139-148. MR 0375448 (51:11641)
  • [3] J. Wolfowitz, Remarks on the notion of recurrence, Bull. Amer. Math. Soc. 55 (1949), 394-395. MR 10, 549. MR 0029109 (10:549e)

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Keywords: Stationary sequences, ergodic theorem, rate of convergence to infinity of partial sums
Article copyright: © Copyright 1975 American Mathematical Society

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