1. Home
  2. Issues
  3. Volume 12, Issue 2 (2025)
  4. Bivariate dependence, stochastic orders ...

Modern Stochastics: Theory and Applications


Bivariate dependence, stochastic orders and conditional tails of the recurrence times in a renewal process
Volume 12, Issue 2 (2025), pp. 153–167
Pub. online: 12 November 2024      Type: Research Article      Open accessOpen Access
Received
14 June 2024
Revised
17 October 2024
Accepted
17 October 2024
Published
12 November 2024

Abstract

The structure of dependence between the forward and the backward recurrence times in a renewal process is considered. Monotonicity properties, as a function of time, for the tail of the bivariate distribution for the recurrence times are discussed, as well as their link with aging properties of the interarrival distribution F. A necessary and sufficient condition for the renewal function to be concave is also obtained. Finally, some properties of the conditional tail for one of the two recurrence times, given some information on the other, are studied. The results are illustrated by some numerical examples.

References

[1] 
Belzunce, F., Martínez–Riquelme, C., Mulero, J.: An Introduction to Stochastic Orders. Academic Press (2016)
[2] 
Belzunce, F., Ortega, E.M., Ruiz, M.: A note on Stochastic Comparison of Excess Lifetimes of Renewal Processes. J. Appl. Probab. 38, 747–753 (2001). https://doi.org/10.1239/jap/1005091037
[3] 
Boutsikas, M.V., Vaggelatou, E.: On the distance between convex-ordered random variables. Adv. Appl. Probab. 34, 349–374 (2002). https://doi.org/10.1239/aap/1025131222
[4] 
Brown, M.: Bounds, inequalities, and monotonicity properties for some specialized renewal processes. Ann. Probab. 8, 227–240 (1980)
[5] 
Chen, Y.-H.: Classes of life distributions and renewal counting processes. J. Appl. Probab. 31, 1110–1115 (1994). https://doi.org/10.2307/3215334
[6] 
Denuit, M., Dhaene, J., Goovaerts, M.J., Kaas, R.: Actuarial Theory for Dependent Risks: Measures, Orders and Models. Wiley, Chichester, UK (2005)
[7] 
Esary, J.D., Proschan, F., Walkup, D.W.: Association of random variables, with applications. Ann. Math. Stat. 44, 1466–1474 (1967). https://doi.org/10.1214/aoms/1177698701
[8] 
Gakis, K.G., Sivazlian, B.D.: The correlation of the backward and forward recurrence times in a renewal process. Stoch. Anal. Appl. 12(5), 543–549 (1994). https://doi.org/10.1080/07362999408809372
[9] 
Gupta, A.K., Zeng, W.B., Wu, Y.: Probability and Statistical Models: Foundations for Problems in Reliability and Financial Mathematics. Birkhäuser, Boston, MA (2010)
[10] 
Janssen, J., Manca, R.: Applied Semi–Markov Processes. Springer, Berlin (2006)
[11] 
Lehmann, E.L.: Some concepts of dependence. Ann. Math. Stat. 37(5), 1137–1153 (1966). https://doi.org/10.1214/aoms/1177699260
[12] 
Losidis, S.: Covariance between the forward recurrence time and the number of renewals. Mod. Stoch. Theory Appl. 9(1), 1–16 (2022)
[13] 
Losidis, S.: Recurrence times and the expected number of renewal epochs over a finite interval. Statistics 57(1), 195–212 (2023). https://doi.org/10.1080/02331888.2023.2172172
[14] 
Losidis, S., Politis, K.: The covariance of the backward and forward recurrence times in a renewal process: the stationary case and asymptotics for the ordinary case. Stoch. Models 35, 51–62 (2019). https://doi.org/10.1080/15326349.2019.1575752
[15] 
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
[16] 
Marshall, A.W., Olkin, I.: Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families. Springer Series in Statistics (2007)
[17] 
Omey, E.A.M., Teugels, J.L.: A note on Blackwell’s renewal theorem. J. Math. Sci. 134, 656–659 (2006). https://doi.org/10.1007/s10958-006-0010-4
[18] 
Polatioglu, H., Sahin, S.: The IFR-DFR property of the forward recurrence-time distribution. IEEE Trans. Reliab. 47, 59–65 (1997). https://doi.org/10.1109/24.690906
[19] 
Resnick, S.: Adventures in Stochastic Processes. Bikhäuser, Boston (1992)
[20] 
Shaked, M., Shanthikumar, J.G.: Stochastic Orders. Springer, New York (2007)
[21] 
Shaked, M., Zhu, H.: Some results on block replacement policies and renewal theory. J. Appl. Probab. 29, 932–946 (1992). https://doi.org/10.2307/3214725
[22] 
Willmot, G.E., Lin, X.: Lundberg Approximations for Compound Distributions with Insurance Applications. Springer, New York (2001)
[23] 
Yu, Y.: Concave renewal functions do not imply DFR interrenewal times. J. Appl. Probab. 48(2), 583–588 (2011). https://doi.org/10.1239/jap/1308662647

Full article Related articles PDF XML
Full article Related articles PDF XML

Copyright
© 2025 The Author(s). Published by VTeX
Open access article under the CC BY license.

Keywords
Renewal process forward recurrence time backward recurrence time aging classes dependence positive regression dependence negative regression dependence upper orthant stochastic order

MSC2020
60K05 60K10

Metrics
since March 2018
435

Article info
views

288

Full article
views

230

PDF
downloads

107

XML
downloads


Share


RSS