The Annals of Probability

Limit Theorems for Delayed Sums

Tze Leung Lai

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Abstract

In this paper, we study analogues of the law of the iterated logarithm for delayed sums of independent random variables. In the i.i.d. case, necessary and sufficient conditions for such analogues are obtained. We apply our results to find convergence rates for expressions of the form $P\lbrack |S_n| > b_n \rbrack$ and $P\lbrack \sup_{k\geqq n} |S_k/b_k| > \varepsilon \rbrack$ for certain upper-class sequences $(b_n)$. In this connection, certain theorems of Erdos, Baum and Katz are also generalized.

Article information

Source
Ann. Probab., Volume 2, Number 3 (1974), 432-440.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996658

Digital Object Identifier
doi:10.1214/aop/1176996658

Mathematical Reviews number (MathSciNet)
MR356193

Zentralblatt MATH identifier
0305.60009

JSTOR
links.jstor.org

Keywords
6030 Delayed first arithmetic means law of the the iterated logarithm Kolmogrovov's exponential bounds rate of convergence upper-class sequences

Citation

Lai, Tze Leung. Limit Theorems for Delayed Sums. Ann. Probab. 2 (1974), no. 3, 432--440. doi:10.1214/aop/1176996658. https://projecteuclid.org/euclid.aop/1176996658


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