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Exact Results and Bounds for the Joint Tail and Moments of the Recurrence Times in a Renewal Process

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Abstract

The best known result about the joint distribution of the backward and forward recurrence times in a renewal process concerns the asymptotic behavior for the tail of that bivariate distribution. In the present paper we study the joint behavior of the recurrence times at a fixed time point t, and we obtain both exact results and bounds for their joint tail behavior. We also obtain results about the joint moments of these two random variables and we show in particular that the expectation of the product between the two recurrence times increases with time when the interarrival distribution has a decreasing failure rate. The results are illustrated by some numerical examples.

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References

  • Barlow RE (1996) Mathematical theory of reliability. SIAM (Classics in Applied Mathematics)

  • Belzunce F, Ortega E M, Ruiz J M (2001) A note on stochastic comparisons of excess lifetimes of renewal processes. J Appl Probab 38:747–753

    Article  MathSciNet  Google Scholar 

  • Boxma O (1984) Joint distribution of sojourn time and queue length in the M/G/1 queue with (in)finite capacity. Eur J Oper Res 16:246–256

    Article  MathSciNet  Google Scholar 

  • Brown M (1980) Bounds, inequalities, and monotonicity properties for some specialised renewal processes. Ann Probab 8:227–240

    Article  MathSciNet  Google Scholar 

  • Brown M (1981) Further Monotonicity Properties for Specialized Renewal Processes. Ann Probab 9:891–895

    Article  MathSciNet  Google Scholar 

  • Chen Y-H (1994) Classes of life distributions and renewal counting processes. J Appl Probab 31:1110–1115

    Article  MathSciNet  Google Scholar 

  • Coleman R (1982) The moments of the forward recurrence time. Eur J Oper Res 9(2):181–183

    Article  MathSciNet  Google Scholar 

  • Constantine AG, Robinson NI (1997) The Weibull renewal function for moderate to large arguments. Computat Statist Data Anal 24:9–27

    Article  MathSciNet  Google Scholar 

  • Feller W (1971) An introduction to probability theory and its applications, vol II, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  • Gakis KG, Sivazlian BD (1992) The use of multiple integrals in the study of the backward and forward recurrence times for the ordinary renewal process. Stoch Anal Appl 10(4):409–416

    Article  MathSciNet  Google Scholar 

  • Gakis K G, Sivazlian B D (1993) A generalization of the inspection paradox in an ordinary renewal process. Stoch Anal Appl 11(1):43–48

    Article  MathSciNet  Google Scholar 

  • Gakis K G, Sivazlian B D (1994) The correlation of the backward and forward recurrence times in a renewal process. Stoch Anal Appl 12(5):543–549

    Article  MathSciNet  Google Scholar 

  • Gupta R C (1984) Some characterizations of renewal densities with emphasis in reliability. Statistics 15:571–579

    MathSciNet  MATH  Google Scholar 

  • Gut A (2009) Stopped random walks: Limit theorems and applications. Springer-Verlag, New York

    Book  Google Scholar 

  • Li X, Kochar S C (2001) Some new results involving the NBU(2) class of life distributions. J Appl Probab 38:242–247

    Article  MathSciNet  Google Scholar 

  • Losidis S, Politis K (2017) A two-sided bound for the renewal function when the interarrival distribution is IMRL. Stat Probab Lett 125:164–170

    Article  MathSciNet  Google Scholar 

  • Losidis S, Politis K (2019) The covariance of the backward and forward recurrence times in a renewal process: the stationary case and asymptotics for the ordinary case. Stoch Model 35:51–62

    Article  MathSciNet  Google Scholar 

  • Psarrakos G, Politis K (2012) The covariance between the surplus prior to and at ruin in the classical risk model. ASTIN Bull 42(2):631–653

    MathSciNet  MATH  Google Scholar 

  • Resnick S (1992) Adventures in stochastic processes. Bikhäuser, Boston

    MATH  Google Scholar 

  • Shaked M, Shanthikumar J G (2007) Stochastic orders. Springer, New York

    Book  Google Scholar 

  • Shaked M, Zhu H (1992) Some results on block replacement policies and renewal theory. J Appl Probab 29:932–946

    Article  MathSciNet  Google Scholar 

  • Willmot GE, Lin X (2001) Lundberg approximations for compound distributions with insurance applications. Springer, New York

    Book  Google Scholar 

Download references

Acknowledgments

The authors wish to thank the referee for their valuable and constructive comments, which improved the presentation of the paper.

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Authors and Affiliations

  1. Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli & Demetriou Str, Piraeus, 18534, Greece

    Sotirios Losidis, Konstadinos Politis & Georgios Psarrakos

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  1. Sotirios Losidis
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  2. Konstadinos Politis
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  3. Georgios Psarrakos
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Correspondence to Konstadinos Politis.

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Losidis, S., Politis, K. & Psarrakos, G. Exact Results and Bounds for the Joint Tail and Moments of the Recurrence Times in a Renewal Process. Methodol Comput Appl Probab 23, 1489–1505 (2021). https://doi.org/10.1007/s11009-020-09787-w

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  • DOI: https://doi.org/10.1007/s11009-020-09787-w

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