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

Abstract

The paper is concerned with the distribution of the levelNof 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 ofNrequires 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.