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On the local behaviour of ladder height distributions

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

J. Bertoin  and
R. A. Doney
J. Bertoin*
Affiliation:
Université Paris VI
R. A. Doney*
Affiliation:
University of Manchester
*
∗ Postal address: Laboratoire de Probabilités, Université Paris VI, t56, 4 Place Jussieu, 75252 Paris, France.
∗∗ Postal address: Statistical Laboratory, Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK.
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Abstract

There is a well-known connection between the asymptotic behaviour of the tail of the distribution of the increasing ladder height and the integrated tail of the step distribution of a random walk which either drifts to –∞, or oscillates and whose decreasing ladder height has finite mean. We establish a similar connection in a local sense; this means that in the lattice case we link the asymptotic behaviours of the mass function of the ladder height distribution and of the tail of the step distribution. We deduce the asymptotic behaviour of the mass function of the maximum of the walk, when this is finite, and also treat the non-lattice case.

Keywords

RANDOM WALKLADDER PHENOMENAMAXIMUMSUB-EXPONENTIALITY

MSC classification

Primary: 60E05: Distributions
Type
Short Communications
Information
Journal of Applied Probability , Volume 31 , Issue 3 , September 1994 , pp. 816 - 821
Copyright
Copyright © Applied Probability Trust 1994 

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References

[1] Bertoin, J. and Doney, R. A. (1994) On conditioning a random walk to stay non-negative. Ann. Prob. To appear.CrossRefGoogle Scholar
[2] Chistyakov, V. P. (1964) A theorem on sums of independent, positive random variables and its applications to branching processes. Theory Prob. Appl. 9, 640–648.Google Scholar
[3] Chover, J., Ney, P. and Wainger, S. (1973) Functions of probability measures. J. Anal. Math. 26, 255–302.Google Scholar
[4] Embrechts, P., Goldie, C. M., and Veraverbeke, N. (1979) Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.Google Scholar
[5] Embrechts, P. and Hawkes, J. (1982) A limit theorem for the tails of discrete infinitely divisible distributions with applications to fluctuation theory. J. Austral. Math. Soc. 32, 412–422.CrossRefGoogle Scholar
[6] Feller, W. E. (1972) An introduction to probability theory and its applications, Vol. II, 2nd edn. Wiley, New York.Google Scholar
[7] Goldie, C. M. (1978) Subexponential distributions and dominated variation tails. J. Appl. Prob. 15, 440–442.Google Scholar
[8] Grübel, R. (1985) Tail-behaviour of ladder height distributions in random walks. J. Appl. Prob. 22, 705–709.Google Scholar
[9] Veraverbeke, N. (1977) Asymptotic behaviour of Wiener-Hopf factors of a random walk. Stoch. Proc. Appl. 5, 27–37.CrossRefGoogle Scholar
[10] Willekens, E. (1987) On the supremum of an infinitely divisible process. Stoch. Proc. Appl. 26, 173–175.CrossRefGoogle Scholar