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A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory

Published online by Cambridge University Press:  09 April 2009

Paul Embrechts  and
John Hawkes
Paul Embrechts
Affiliation:
Departement Wiskunde KUL, Celestijnenlaan 200-B, B-3030 Heverlee, Belgium
John Hawkes
Affiliation:
University College of Swansea, Department of Statistics, Singleton Park, Swansea SA2 8PP, United Kingdom
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Abstract

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Suppose that (pn) is an infinitely divisible distribution on the non-negative integers having Lévy measure (vn). In this paper we derive a necessary and sufficient condition for the existence of the limit limn→∞ pn/vn. We also derive some other results on the asymptotic behaviour of the sequence (Pn) and apply some of our results to the theory of fluctuations of random walks. We obtain a necessary and sufficient condition for the first positive ladder epoch to belong to the domain of attraction of a spectrally positive stable law with index α, α ∈ (1,2).

Keywords

infinitely divisible sequencesLévy measuredomain of attractionfluctuation theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 32 , Issue 3 , June 1982 , pp. 412 - 422
Copyright
Copyright © Australian Mathematical Society 1982

References

Borovkov, A. A. (1976), Stochastic processes in queueing theory (Springer-Verlag, New York).Google Scholar
Chover, J., Ney, P. and Wainger, S. (1973a), ‘Functions of probability measures’, J. Analyse Math. 26, 255–302.CrossRefGoogle Scholar
Chover, J., Ney, P. and Wainger, S. (1973b), ‘Degeneracy properties of subcritical branching processes’, Ann. Probability 1, 663–673.CrossRefGoogle Scholar
Cohen, J. W. (1973), ‘Some results on regular variation for distributions in queueing and fluctuation theory’, J. Appl. Probability 10, 343–353.CrossRefGoogle Scholar
Doney, R. A. (1977), ‘A note on a condition satisfied by certain random walks’, J. Appl. Probability 14, 843–849.Google Scholar
Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979), ‘Subexponentiality and infinite divisibility’, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 49, 335–347.Google Scholar
Feller, W. (1971), An introduction to probability theory and is applications. Volume 1 (second edition) (Wiley, New York).Google Scholar
Hawkes, J. and Jenkins, J. D. (1978), ‘Infinitely divisible sequences’, Scand. Actuar. J. 2, 65–76.CrossRefGoogle Scholar
Nagaev, S. V. (1979), ‘Large deviations of sums of independent random variables’, Ann. Probability 7, 745–789.CrossRefGoogle Scholar
Pakes, A. G. (1975), ‘On the tails of waiting time distributions’, J. Appl. Probability 12, 555–564.CrossRefGoogle Scholar
Rogozin, B. A. (1971), ‘The distribution of the first ladder moment and height and fluctuation of a random walk’, Theor. Probability Appl. 16, 575–595.CrossRefGoogle Scholar
Rudin, W. (1973), ‘Limits of ratios of tails of measures’, Ann. Probability 1, 982–994.Google Scholar
Spitzer, F. (1956), ‘A combinatorial lemma and its applications to probability theory’, Trans. Amer. Math. Soc. 82, 323–338.CrossRefGoogle Scholar
Teugels, J. L. (1975), ‘The class of subexponential distributions’, Ann. Probability 3, 1000–1011.CrossRefGoogle Scholar
Wright, E. M. (1967), ‘A relationship between two sequences’, Proc. London Math. Soc. 17, 296–304.CrossRefGoogle Scholar