## The Annals of Probability

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

### Limit Theorems for Delayed Sums

#### 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