Hostname: page-component-6898cfcb55-th4rj Total loading time: 0 Render date: 2026-11-12T01:52:06.211Z Has data issue: false hasContentIssue false
  • English
  • Français

Transient renewal processes in the subexponential case

Published online by Cambridge University Press:  14 July 2016

Emily S. Murphree
Emily S. Murphree*
Affiliation:
Miami University
*
∗Postal address: Department of Mathematics and Statistics, Miami University, Bachelor Hall, Oxford, OH 45056, USA.
Get accessRights & Permissions [Opens in a new window]

Abstract

A transient renewal process based on a sequence of possibly infinite waiting times is defined. The process is studied when the (rescaled) distribution of the waiting times belongs to the subexponential class of distributions. In this case, even conditional on all waiting times observed by time t being finite, the distributions of the forward and backward delays at t are asymptotically degenerate. Also, the conditional moments of the number of events by time t converge to the same finite limits as the unconditional moments.

Keywords

FORWARD DELAYBACKWARD DELAY

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 1 , March 1987, pp. 88 - 96
Copyright
Copyright © Applied Probability Trust 1987 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, New York.10.1007/978-3-642-65371-1Google Scholar
Chistyakov, V. P. (1964) A theorem on sums of independent positive random variables. Theory Prob. Appl. 9, 640–648.10.1137/1109088Google Scholar
Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979) Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.10.1007/BF00535504Google Scholar
Murphree, E. S. (1981) Transient cumulative processes. Institute of Statistics Mimeo Series No. 1367, University of North Carolina at Chapel Hill.Google Scholar
Murphree, E. S. and Smith, W. L. (1986) On transient regenerative processes. J. Appl. Prob. 23, 52–70.10.2307/3214116Google Scholar
Pitman, E. J. G. (1980) Subexponential distribution functions. J. Austral. Math. Soc. A 29, 337–347.10.1017/S1446788700021340Google Scholar
Smith, W. L. (1959) On the cumulants of renewal processes. Biometrika 46, 1–29.Google Scholar
Smith, W. L. (1972) On the tails of queueing-time distributions. Institute of Statistics Mimeo Series No. 830, University of North Carolina at Chapel Hill.Google Scholar
Teugels, J. (1975) the class of subexponential distributions. Ann. Prob. 3, 1000–1011.10.1214/aop/1176996225Google Scholar