Hostname: page-component-6bf8c574d5-t27h7 Total loading time: 0 Render date: 2025-02-21T21:07:42.605Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A GENERALIZATION OF THE STATIONARY EXCESS OPERATOR

Published online by Cambridge University Press:  30 March 2015

Yoav Kerner  and
Andreas Löpker
Yoav Kerner
Affiliation:
Ben-Gurion University of the Negev, P.O.B 653, Beer-Sheva 84105, Israel E-mail: [email protected]
Andreas Löpker
Affiliation:
Helmut Schmidt University Hamburg, 22043 Hamburg, Germany E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that overshoots over Erlang random variables give rise to a natural generalization of the stationary excess operator and its iterates. The new operators can be used to derive expansions for the expectation Eg(X) of a non-negative random variable, similar to Taylor-like expansions encountered when studying stationary excess operators.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 29 , Issue 2 , April 2015 , pp. 219 - 232
Copyright
Copyright © Cambridge University Press 2015 

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.)

References

1.Abate, J. & Whitt, W. (1996). An operational calculus for probability distributions via Laplace transforms. Advances in Applied Probability 28(1): 75–113.CrossRefGoogle Scholar
2.Asmussen, S. (2003). Applied Probability and Queues, Applications of Mathematics 51. 2nd edn., Springer.Google Scholar
3.Brown, M. (2006). Exploiting the waiting time paradox. Applications of the size-biasing transformation. Probability in the Engineering and Informational Sciences 20(2): 195–230.Google Scholar
4.Cox, D.R. (1962). Renewal Theory, John Wiley & Sons.Google Scholar
5.Denuit, M. (2002). S-convex extrema, Taylor-type expansions and stochastic approximations. Scandinavian Actuarial Journal 2002(1): 45–67.Google Scholar
6.Doob, J.L. (1948). Renewal theory from the point of view of the theory of probability. Transactions of the American Mathematical Society 63: 422–438.Google Scholar
7.Ferrari, P.A., Kesten, H., Martinez, S. & Picco, P. (1995). Existence of quasi-stationary distributions: a renewal dynamical approach. Annals of Probability 23(2): 501–521.Google Scholar
8.Gupta, R.C. (2007). Role of equilibrium distribution in reliability studies. Probability in the Engineering Informational Sciences 21(2): 315–334.Google Scholar
9.Harkness, W.L. & Shantaram, R. (1969). Convergence of a sequence of transformations of distribution functions. Pacific Mathematical Journal 31: 403–415.Google Scholar
10.Hesselager, O., Wang, Sh. & Willmot, G.E. (1998). Exponential and scale mixtures and equilibrium distributions. Scandinavian Actuarial Journal. 1998(2): 125–142.Google Scholar
11.Huang, J.S. & Lin, G.D. (1995). Characterization of distributions using equilibrium transformations. Sankhyā, Series A 57(2): 179–185.Google Scholar
12.Lin, G.D. (1994). On a probabilistic generalization of Taylor's theorem. Statistics & Probability Letters 19(3): 237–243.Google Scholar
13.Massey, W.A. & Whitt, W. (1993). A probabilistic generalization of Taylor's theorem. Statistics & Probability Letters 16(1): 51–54.Google Scholar
14.Nanda, A.K., Jain, K. & Singh, H. (1996). On closure of some partial orderings under mixtures. Journal of Applied Probability 33(3): 698–706.CrossRefGoogle Scholar
15.Pakes, A.G. (1996). Length biasing and laws equivalent to the log-normal. Journal of Mathematical Analysis and Applications 197(3): 825–854.Google Scholar
16.Pakes, A.G. (2004). Characterization of probability laws via length-biasing and stationary-excess operations. Communications in Statistics, Theory and Methods 33(11–12): 2975–2992.Google Scholar
17.Shantaram, R. & Harkness, W. (1972). On a certain class of limit distributions. Annals of Mathematics and Statistics 43: 2067–2071.Google Scholar
18.van Beek, P. & Braat, J. (1973). The limits of sequences of iterated overshoot distribution functions. Stochastic Processes and their Applications 1: 307–316.Google Scholar
19.Vardi, Y., Shepp, L.A. & Logan, B.F. (1981). Distribution functions invariant under residual-lifetime and length biased sampling. Zeitschrift fr scheinlichkeitstheorie und Verwandte Gebiete 56: 415–426.Google Scholar
20.Whitt, W. (1985). The renewal-process stationary-excess operator. Journal of Applied Probability 22: 156–167.Google Scholar