Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-13T03:57:19.318Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for general shock models with infinite mean intershock times

Published online by Cambridge University Press:  14 July 2016

Kevin K. Anderson
Kevin K. Anderson*
Affiliation:
IBM T. J. Watson Research Center
Get access Rights & Permissions [Opens in a new window]

Abstract

A general shock model in which the time intervals between shocks have infinite expectation is considered. Limit theorems for the first time the magnitude of a shock exceeds a and the historical maximum magnitude are given.

Keywords

EXTREME VALUE THEORYRENEWAL THEORYREGULAR VARIATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 2 , June 1987 , pp. 449 - 456
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.)

References

Anderson, K. K. (1985) Limit Theorems for Renewal Processes in the Infinite Mean Case. Ph.D. Dissertation, Iowa State University.Google Scholar
Barndorff-Nielsen, O. (1964) On the limit distribution of the maximum of a random number of independent random variables. Acta Math. Acad. Sci. Hung. 15, 399–403.Google Scholar
Berman, S. M. (1962) Limiting distribution of the maximum term in sequences of dependent random variables. Ann. Math. Statist. 33, 894–908.Google Scholar
Darling, D. A. and Kac, M. (1957) On occupation times for Markoff processes. Trans. Amer. Math. Soc. 84, 444–458.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd ed. Wiley, New York.Google Scholar
Galambos, J. (1978) The Asymptotic Theory of Extreme Order Statistics. Wiley, New York.Google Scholar
Gaver, D. P. (1976) Random record models. J. Appl. Prob. 13, 538–547.Google Scholar
Gnedenko, B. V. (1972) Limit theorems for sums of a random number of positive independent random variables. Proc. 6th Berkeley Symp. Math. Statist. Prob. 2, 537–549.Google Scholar
Rogozin, B. A. (1971) The distribution of the first ladder moment and height and fluctuation of a random walk. Theory Prob. Appl. 16, 575–595.Google Scholar
Seneta, E. (1976) Regularly Varying Functions. Lecture Notes in Mathematics 508, Springer-Verlag, Berlin.Google Scholar
Shanthikumar, J. G. and Sumita, U. (1983) General shock models associated with correlated renewal sequences. J. Appl. Prob. 20, 600–614.Google Scholar
Westcott, M. (1977) The random record model. Proc. R. Soc. London A 356, 529–547.Google Scholar