## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 2, Number 3 (1992), 714-738.

### On Moments of the First Ladder Height of Random Walks with Small Drift

#### Abstract

This paper presents some results that are useful in the study of asymptotic approximations of boundary crossing probabilities for random walks. The main result is a refinement of an asymptotic expansion of Siegmund concerning moments of the first ladder height of random walks having small positive drift. An analysis of the covariance between the first passage time and the overshoot of a random walk over a horizontal boundary contributes to the development of the main result and is of independent interest as well. An application of these results to a "moderate deviations" approximation for the probability distribution of the time to false alarm in the cusum procedure is briefly described.

#### Article information

**Source**

Ann. Appl. Probab., Volume 2, Number 3 (1992), 714-738.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005656

**Digital Object Identifier**

doi:10.1214/aoap/1177005656

**Mathematical Reviews number (MathSciNet)**

MR1177906

**Zentralblatt MATH identifier**

0760.60064

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J15

Secondary: 60F99: None of the above, but in this section 62L10: Sequential analysis

**Keywords**

Random walk exponential family uniform renewal theorem first ladder height first passage time overshoot boundary crossing probability cusum procedure corrected diffusion approximation moderate deviations

#### Citation

Chang, Joseph T. On Moments of the First Ladder Height of Random Walks with Small Drift. Ann. Appl. Probab. 2 (1992), no. 3, 714--738. doi:10.1214/aoap/1177005656. https://projecteuclid.org/euclid.aoap/1177005656