## The Annals of Probability

- Ann. Probab.
- Volume 4, Number 4 (1976), 628-643.

### Uniform Tauberian Theorems and their Applications to Renewal Theory and First Passage Problems

#### Abstract

In this paper, we prove an analogue of the classical renewal theorem for the case where there is no drift. Our proof depends on a uniform version of Spitzer's well-known theorem on ladder epochs and ladder variables, and we obtain this uniform result by using uniform Tauberian theorems. Some further applications of these uniform Tauberian theorems to other problems in renewal theory and first passage times are also given.

#### Article information

**Source**

Ann. Probab., Volume 4, Number 4 (1976), 628-643.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996032

**Digital Object Identifier**

doi:10.1214/aop/1176996032

**Mathematical Reviews number (MathSciNet)**

MR410966

**Zentralblatt MATH identifier**

0365.60095

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F99: None of the above, but in this section

Secondary: 60K05: Renewal theory

**Keywords**

Renewal theory first passage problems ladder epoch ladder variable uniform Tauberian theorems uniform strong law of large numbers Paley-type inequalities

#### Citation

Lai, Tze Leung. Uniform Tauberian Theorems and their Applications to Renewal Theory and First Passage Problems. Ann. Probab. 4 (1976), no. 4, 628--643. doi:10.1214/aop/1176996032. https://projecteuclid.org/euclid.aop/1176996032