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May, 1994 Quadrature Routines for Ladder Variables
Robert W. Keener
Ann. Appl. Probab. 4(2): 570-590 (May, 1994). DOI: 10.1214/aoap/1177005073


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.


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Robert W. Keener. "Quadrature Routines for Ladder Variables." Ann. Appl. Probab. 4 (2) 570 - 590, May, 1994.


Published: May, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0803.60065
MathSciNet: MR1272740
Digital Object Identifier: 10.1214/aoap/1177005073

Primary: 60J15
Secondary: 60E10

Keywords: Hilbert transform , nonlinear renewal theory , Random walks

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.4 • No. 2 • May, 1994
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