The Annals of Applied Probability

Quadrature Routines for Ladder Variables

Robert W. Keener

Full-text: Open access

Abstract

Let $T = \inf\{n \geq 1: S_n > 0\}$ and $H = S_T$ be ladder variables for a random walk $\{S_n\}_{n \geq 1}$ with nonnegative drift. Integral formulas for generating functions and moments of $T, H$ and related quantities are developed. These formulas are suitable for numerical quadrature and should be easier to implement than formulas based on Spitzer's identity when the distribution of $S_n$ is complicated. The approach used makes key use of the Hilbert transform and the main regularity assumption is that some power of the characteristic function for steps of the random walk is integrable.

Article information

Source
Ann. Appl. Probab., Volume 4, Number 2 (1994), 570-590.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1177005073

Digital Object Identifier
doi:10.1214/aoap/1177005073

Mathematical Reviews number (MathSciNet)
MR1272740

Zentralblatt MATH identifier
0803.60065

JSTOR
links.jstor.org

Subjects
Primary: 60J15
Secondary: 60E10: Characteristic functions; other transforms

Keywords
Random walks Hilbert transform nonlinear renewal theory

Citation

Keener, Robert W. Quadrature Routines for Ladder Variables. Ann. Appl. Probab. 4 (1994), no. 2, 570--590. doi:10.1214/aoap/1177005073. https://projecteuclid.org/euclid.aoap/1177005073


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