Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T16:32:33.604Z Has data issue: false hasContentIssue false
  • English
  • Français

First crossing of basic counting processes with lower non-linear boundaries: A unified approach through pseudopolynomials (I)

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Philippe Picard  and
Claude Lefèvre
Philippe Picard*
Affiliation:
Université de Lyon 1
Claude Lefèvre*
Affiliation:
Université Libre de Bruxelles
*
∗ Postal address: Mathématiques Appliquées, Université de Lyon 1, 43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cede x, France.
∗∗ Postal address: Institut de Statistique, C.P.210, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Bruxelles, Belgique.
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper is concerned with the distribution of the level N of the first crossing of a counting process trajectory with a lower boundary. Compound and simple Poisson or binomial processes, gamma renewal processes, and finally birth processes are considered. In the simple Poisson case, expressing the exact distribution of N requires the use of a classical family of Abel–Gontcharoff polynomials. For other cases convenient extensions of these polynomials into pseudopolynomials with a similar structure are necessary. Such extensions being applicable to other fields of applied probability, the central part of the present paper has been devoted to the building of these pseudopolynomials in a rather general framework.

Keywords

LOWER AND UPPER BOUNDARIESABEL–GONTCHAROFF POLYNOMIALSAPPELL POLYNOMIALS

MSC classification

Primary: 60G17: Sample path properties
Secondary: 60J75: Jump processes 60G40: Stopping times; optimal stopping problems; gambling theory
Type
General Applied Probablity
Information
Advances in Applied Probability , Volume 28 , Issue 3 , September 1996 , pp. 853 - 876
Copyright
Copyright © Applied Probability Trust 1996 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research partially supported by the Institut National de la Santé et de la Recherche Médicale under contract No. 921011.

References

Abel, N. (1881) Sur les fonctions génératrices et leurs déterminantes. Oeuvres Complètes. Vol 2. pp. 67–81.Google Scholar
Appell, P. E. (1880) Sur une classe de polynômes. Ann. Sci. Ecole Normale Sup. 9, 119–144.CrossRefGoogle Scholar
Boas, R. P. Jr. and Buck, R. C. (1958) Polynomial Expansions of Analytic Functions. Springer, Heidelberg.Google Scholar
Consul, P. C. (1989) Generalized Poisson Distribution. Properties and Applications . Marcel Dekker, New York.Google Scholar
Daniels, H. E. (1963) The Poisson process with a curved absorbing boundary. Bull. Int. Statist. Inst., 34th session. 40, 994–1008.Google Scholar
Daniels, H. E. (1967) The distribution of the total size of an epidemic. Proc. 5th Berkeley Symp. Math. Statist. Prob. 4, 281–293.Google Scholar
Gallot, S. F. L. (1966) Asymptotic absorption probabilities for a Poisson process. J. Appl. Prob. 3, 445–452.CrossRefGoogle Scholar
Gallot, S. F. L. (1993) Absorption and first passage times for a compound Poisson process in a general upper boundary. J. Appl. Prob. 30, 835–850.Google Scholar
Gontcharoff, W. (1930) Recherches sur les dérivées successives des fonctions analytiques. Généralisation de la série d'Abel. Ann. Sci. Ecole Normale Sup. 47, 1–78.CrossRefGoogle Scholar
Kaz'Min, Y. A. (1988) Appell polynomials. In Encyclopedia of Mathematics , ed. Hazewinkel, M.. Kluwer, Dordrecht. pp. 1, 209–210.Google Scholar
Lefevre, Cl. and Picard, Ph. (1989) Quelques nouvelles classes de distributions engendrées par des familles de polynômes non-standards. Bull. Int. Statist. Inst., 47th session. 2, 32–33.Google Scholar
Lefevre, Cl. and Picard, Ph. (1990) A non-standard family of polynomials and the final size distribution of Reed–Frost epidemic processes. Adv. Appl. Prob. 22, 25–48.Google Scholar
Lefevre, Cl. and Picard, Ph. (1996) Abelian-type expansions and non-linear death processes (II). Adv. Appl. Prob. 28, 877–894.Google Scholar
Mullin, R. and Rota, G.-C. On the foundations of combinatorial theory—III: Theory of binomial enumeration. In Graph Theory and its Applications. Academic Press, New York. pp 167–213.Google Scholar
Niederhausen, H. (1981) Sheffer polynomials for computing exact Kolmogorov–Smimov and Rényi type distributions. Ann. Statist. 9, 923–944.CrossRefGoogle Scholar
Niederhausen, H. (1988) Sheffer polynomials. In Encyclopedia of Statistical Sciences. ed. Kotz, S., Johnson, N. and Read, C.. Wiley, New York. pp 8, 436–441.Google Scholar
Picard, Ph. (1980) Applications of martingale theory to some epidemic models. J. Appl. Prob. 17, 583–599.Google Scholar
Picard, Ph. and Lefevre, Cl. (1989) Sur des familles de polynômes non-standards intervenant dans des problèmes de premier passage. Bull. Int. Statist. Inst., 47th session. 2, 208–209.Google Scholar
Picard, Ph. and Lefevre, Cl. (1990) A unified analysis of the final size and severity distribution in collective Reed–Frost epidemic processes. Adv. Appl. Prob. 22, 269–294.Google Scholar
Picard, Ph. and Lefevre, Cl. (1994) On the first crossing of the surplus process with a given upper barrier. Insurance: Math. Econ. 14, 163–179.Google Scholar
Schäl, M. (1993) On hitting times for jump-diffusion processes with past dependent local characteristics. Stoch. Proc. Appl. 47, 131–142.Google Scholar
Sheffer, I. M. (1939) Some properties of polynomial sets of type zero. Duke Math. J. 5, 590–622.Google Scholar
Stadje, W. (1993) Distribution of first-exit times for empirical counting and Poisson processes with moving boundaries. Commun. Statist. Stoch. Models 9, 91–103.Google Scholar
Whittle, P. (1961) Some exact results for one-sided distribution tests of the Kolmogorov-Smimov type. Ann. Math. Statist. 32, 499–505.Google Scholar
Zacks, S. (1991) Distributions of stopping times for Poisson processes with linear boundaries. Commun. Statist. Stoch. Models 7, 233–242.CrossRefGoogle Scholar