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

Author

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  • Sotirios Losidis

    (University of Piraeus)

  • Konstadinos Politis

    (University of Piraeus)

  • Georgios Psarrakos

    (University of Piraeus)

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.

Suggested Citation

  • Sotirios Losidis & Konstadinos Politis & Georgios Psarrakos, 2021. "Exact Results and Bounds for the Joint Tail and Moments of the Recurrence Times in a Renewal Process," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1489-1505, December.
    • Handle: RePEc:spr:metcap:v:23:y:2021:i:4:d:10.1007_s11009-020-09787-w
    DOI: 10.1007/s11009-020-09787-w
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    References listed on IDEAS

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    1. Constantine, A. G. & Robinson, N. I., 1997. "The Weibull renewal function for moderate to large arguments," Computational Statistics & Data Analysis, Elsevier, vol. 24(1), pages 9-27, March.
    2. Psarrakos, Georgios & Politis, Konstadinos, 2012. "The Covariance Between the Surplus Prior to and at Ruin in the Classical Risk Model," ASTIN Bulletin, Cambridge University Press, vol. 42(2), pages 631-653, November.
    3. Dickson, D. C. M., 2001. "Lundberg Approximations for Compound Distributions with Insurance Applications. By G. E. Willmot and X. S. Lin. (Springer, 2000)," British Actuarial Journal, Cambridge University Press, vol. 7(4), pages 690-691, October.
    4. Boxma, O. J., 1984. "Joint distribution of sojourn time and queue length in the M/G/1 queue with (in) finite capacity," European Journal of Operational Research, Elsevier, vol. 16(2), pages 246-256, May.
    5. Losidis, Sotirios & Politis, Konstadinos, 2017. "A two-sided bound for the renewal function when the interarrival distribution is IMRL," Statistics & Probability Letters, Elsevier, vol. 125(C), pages 164-170.
    6. Coleman, Rodney, 1982. "The moments of forward recurrence time," European Journal of Operational Research, Elsevier, vol. 9(2), pages 181-183, February.
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