Hostname: page-component-7479d7b7d-c9gpj Total loading time: 0 Render date: 2024-07-13T10:34:20.316Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic models as functionals: some remarks on the renewal case

Published online by Cambridge University Press:  14 July 2016

Rudolf Grübel
Rudolf Grübel*
Affiliation:
Imperial College, London
*
∗ Postal address: Department of Mathematics, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

We regard the standard stochastic model of renewal theory as a functional which associates with a lifetime distribution μ the corresponding renewal measure v = Σ0∞μ ∗n. The behaviour of this functional near exponential distributions is investigated. The literature mainly deals with the asymptotic behaviour of v([0, t]) and v[t, t + 1]) as t → ∞— this new approach leads to expansions which are valid uniformly in t ∈ R.

Keywords

RENEWAL THEORYLIFETIME DISTRIBUTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 2 , June 1989 , pp. 296 - 303
Copyright
Copyright © Applied Probability Trust 1989 

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

Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
Carlsson, H. (1983) Remainder term estimates of the renewal function. Ann. Prob. 11, 143–157.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, II. Wiley, New York.Google Scholar
Grübel, R. (1982) Über das asymptotische Verhalten von Erneuerungsdichten. Math. Nachrichten 108, 39–48.Google Scholar
Grübel, R. (1987) On subordinated distributions and generalized renewal measures. Ann. Prob. 15, 394–415.Google Scholar
Pang, S. K. W. (1987) Alternating Renewal Processes. M.Sc. Thesis, Imperial College, London.Google Scholar
Rudin, W. (1974) Functional Analysis. Tata McGraw-Hill, New Delhi.Google Scholar