The Annals of Applied Probability

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

Joseph T. Chang

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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

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

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60J15
Secondary: 60F99: None of the above, but in this section 62L10: Sequential analysis

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


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.

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