• Bernoulli
  • Volume 23, Number 4B (2017), 3508-3536.

Asymptotic expansions and hazard rates for compound and first-passage distributions

Ronald W. Butler

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A general theory which provides asymptotic tail expansions for density, survival, and hazard rate functions is developed for both absolutely continuous and integer-valued distributions. The expansions make use of Tauberian theorems which apply to moment generating functions (MGFs) with boundary singularities that are of gamma-type or log-type. Standard Tauberian theorems from Feller [An Introduction to Probability Theory and Its Applications II (1971) Wiley] can provide a limited theory but these theorems do not suffice in providing a complete theory as they are not capable of explaining tail behaviour for compound distributions and other complicated distributions which arise in stochastic modelling settings. Obtaining such a complete theory for absolutely continuous distributions requires introducing new “Ikehara” conditions based upon Tauberian theorems whose development and application have been largely confined to analytic number theory. For integer-valued distributions, a complete theory is developed by applying Darboux’s theorem used in analytic combinatorics. Characterizations of asymptotic hazard rates for both absolutely continuous and integer-valued distributions are developed in conjunction with these expansions. The main applications include the ruin distribution in the Cramér–Lundberg and Sparre Andersen models, more general classes of compound distributions, and first-passage distributions in finite-state semi-Markov processes. Such first-passage distributions are shown to have exponential-like/geometric-like tails which mimic the behaviour of first-passage distributions in Markov processes even though the holding-time MGFs involved with such semi-Markov processes are typically not rational.

Article information

Bernoulli, Volume 23, Number 4B (2017), 3508-3536.

Received: February 2015
Revised: December 2015
First available in Project Euclid: 23 May 2017

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asymptotic hazard rate compound distribution Cramér–Lundberg approximation Darboux’s theorem first-passage distribution Ikehara–Delange theorem Ikehara–Wiener theorem semi-Markov process Sparre Andersen model Tauberian theory


Butler, Ronald W. Asymptotic expansions and hazard rates for compound and first-passage distributions. Bernoulli 23 (2017), no. 4B, 3508--3536. doi:10.3150/16-BEJ854.

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Supplemental materials

  • Supplement to “Asymptotic expansions and hazard rates for compound and first-passage distributions”. The Appendices can be found in the supplementary material referenced as [11] below. This material is comprised of two Appendices. APPENDIX A: Contains proofs for asymptotic hazards, proofs using Feller conditions, examples with branch-point singularities, proofs for integer-valued distributions, and proofs with logarithmic singularities. APPENDIX B: Contains proofs using Ikehara conditions and all derivations for compound distributions and first-passage distributions in SMPs.