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Distribution theory for dependent renewal–reward processes and their first-passage times using saddlepoint and residue expansions

Part of: Special processes

Published online by Cambridge University Press:  05 December 2024

Ronald W. Butler
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Abstract

The distribution theory for discrete-time renewal–reward processes with dependent rewards is developed through the derivation of double transforms. By dependent, we mean the more realistic setting in which the reward for an interarrival period is dependent on the duration of the associated interarrival time. The double transforms are the generating functions in time of the time-dependent reward probability-generating functions. Residue and saddlepoint approximations are used to invert such double transforms so that the reward distribution at arbitrary time n can be accurately approximated. In addition, double transforms are developed for the first-passage time distribution that the cumulative reward exceeds a fixed threshold amount. These distributions are accurately approximated by inverting the double transforms using residue and saddlepoint approximation methods. The residue methods also provide asymptotic expansions for moments and allow for the proof of central limit theorems related to these first passage times and reward amounts.

Keywords

Compound renewal processescumulative processesdouble transformsrenewal–reward processesresidue approximationssaddlepoint approximationsshock models

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 2 , June 2025 , pp. 447 - 474
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Borovkov, A. A. (1965). New limit theorems in boundary problems for sums of independent terms. In Selected Translations in Mathematical Statistics and Probability, Vol. 5. American Mathematical Society, Providence, RI, pp. 315–372.Google Scholar
Borovkov, A. A and Rogozin, B. A. (1964). Boundary problems in some two-dimensional random walks. Theory Prob. Appl. 9, 361–388.CrossRefGoogle Scholar
Butler, R. W. (2007). Saddlepoint Approximations with Applications. Cambridge University Press.CrossRefGoogle Scholar
Butler, R. W. (2019). Asymptotic expansions and saddlepoint approximations using the analytic continuation of moment generating functions. J. Appl. Prob. 56, 307–338.CrossRefGoogle Scholar
Butler, R. W. (2023). Residue expansions and saddlepoint approximations in stochastic models using the analytic continuation of generating functions. Stoch. Models 39, 469–501.CrossRefGoogle Scholar
Butler, R. W. (2024). Approximate double-transform inversion when time is one of the variables. Bernoulli 30, 2878–2903.CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn. John Wiley, New York.Google Scholar
Howard, R. A. (1971). Dynamic Probabilistic Systems, Vol. I, Markov Models. John Wiley, New York.Google Scholar
Johnson, N. J., Kotz, S. and Balakrishnan, N. (1997). Discrete Multivariate Distributions. John Wiley, New York.Google Scholar
Kao, E. P. C. (1997). An Introduction to Stochastic Processes. Wadsworth Publishing Company, Belmont, CA.Google Scholar
Kocherlakota, S. and Kocherlakota, K. (1992). Bivariate Discrete Distributions. Marcel Dekker, New York.Google Scholar
Kulkarni, V. G. (2017). Modeling and Analysis of Stochastic Systems, 3rd edn. CRC Press, New York.Google Scholar
Lotov, V. I. (1980). Asymptotic analysis of distributions in problems with two boundaries. I. Theory Prob. Appl. 24, 480–491.CrossRefGoogle Scholar
Lotov, V. I. (1980). Asymptotic analysis of distributions in problems with two boundaries. II. Theory Prob. Appl. 24, 869–876.CrossRefGoogle Scholar
Lugannani, R. and Rice, S. O. (1980). Saddlepoint approximations for the distribution of the sum of independent random variables. Adv. Appl. Prob. 12, 475–490.CrossRefGoogle Scholar
Nakagawa, T. (2011). Stochastic Processes with Applications to Reliability Theory. Springer, New York.CrossRefGoogle Scholar
Presman, È. L. (1969). Factorization methods and boundary problems for sums of random variables given on Markov chains. Math. USSR-Izv. 3, 815–852.CrossRefGoogle Scholar
Skovgaard, I. M. (1987). Saddlepoint expansions for conditional distributions. J. Appl. Prob. 24, 875–887.CrossRefGoogle Scholar
Tijms, H. C. (2003). A First Course in Stochastic Models. John Wiley, New York.CrossRefGoogle Scholar
Zhao, X. and Nakagawa, T. (2018). Advanced Maintenance Policies for Shock and Damage Models. Springer, New York.CrossRefGoogle Scholar
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