Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T16:57:55.222Z Has data issue: false hasContentIssue false
  • English
  • Français

Inverses of extremal processes

Published online by Cambridge University Press:  01 July 2016

Sidney I. Resnick
Sidney I. Resnick*
Affiliation:
Stanford University
Get access Rights & Permissions [Opens in a new window]

Abstract

The inverse of an extremal process {Y(t), t ≧ 0} is an additive process whose Lévy measure can be computed. This measure controls among other things the Poisson number of jumps of Y while Y is in the vertical window (c, d]. A simple transformation of the inverse of the extremal process governed by Λ (x) = exp{– e–x} is also extremal-Λ (x) and this fact enables one to relate behavior of Y-Λ at t = ∞ to behavior near t = 0. Some extensions of these ideas to sample sequences of maxima of i.i.d. random variables are carried out.

Keywords

EXTREMAL PROCESSESMAXIMAITERATED LOGARITHMADDITIVE PROCESSESRECORD VALUESINVERSES
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 2 , June 1974 , pp. 392 - 406
Copyright
Copyright © Applied Probability Trust 1974 

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] Barndorff-Nielsen, O. (1961) On the rate of growth of the partial maxima of a sequence of independent identically distributed random variables. Math. Scand. 9, 383–384.CrossRefGoogle Scholar
[2] De Haan, L. (1970) On Regular Variation and its Application to the Weak Convergence. of Sample Extremes. MC Tract 32, Mathematisch Centrum, Amsterdam.Google Scholar
[3] De Haan, L. and Hordijk, A. (1972) The rate of growth of sample maxima. Ann. Math. Statist. 43, 1185–1196.CrossRefGoogle Scholar
[4] De Haan, L. and Resnick, S. I. (1973) Almost sure limit points of record values J. Appl. Prob. 10, 528–542.Google Scholar
[5] De Haan, L. (1972) Equivalence classes of regularly varying functions. Technical Report No. 41, Department of Statistics, Stanford University.Google Scholar
[6] Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
[7] Dwass, M. (1964) Extremal processes. Ann. Math. Statist. 35, 1718–1725.CrossRefGoogle Scholar
[8] Dwass, M. (1966) Extremal processes II. Illinois J. Math. 10, 381–395.CrossRefGoogle Scholar
[9] Dwass, M. (1973) Extremal processes III. Discussion Paper No. 41, Center for Mathematical Studies in Economics and Management Science, Northwestern University.Google Scholar
[10] Feller, W. (1971) An Introduction to Probability Theory and its Applications. Vol. 2, 2nd ed. Wiley, New York.Google Scholar
[11] Gikhman, I. I. and Skorokhod, A. V. (1969) Introduction to the Theory of Random Processes. Saunders, London.Google Scholar
[12] Îto, K. (1969) Stochastic Processes. Lecture Notes Series No. 16, Matematisk Institut, Aarhus Universitet.Google Scholar
[13] Resnick, S. I. and Rubinovitch, M. (1973) The structure of extremal processes. Adv. Appl. Prob. 5, 287–308.Google Scholar
[14] Resnick, S. I. (1973) Extremal processes and record value times. J. Appl. Prob. 10, 864–868.Google Scholar
[15] Resnick, S. I. (1973) Weak convergence to extremal processes. Technical Report No. 49, Dept. of Statistics, Stanford University. Ann. Probab. To appear.Google Scholar
[16] Robbins, H. and Siegmund, D. (1971) On the law of the iterated logarithm for maxima and minima. Proceedings of the Sixth Berkeley Symposium, Berkeley, California.Google Scholar
[17] Shorrock, R. (1972) On record values and record times. J. Appl. Prob. 9, 316–326.Google Scholar
[18] Shorrock, R. (1973) Record values and inter-record times. J. Appl. Prob. 10, 543–555.Google Scholar
[19] Shorrock, R. (1974) On discrete time extremal processes. J. Appl. Prob. 11, No. 3. To appear.Google Scholar
[20] Vervaat, W. (1973) Limit theorems for partial maxima and records. Preprint , Dept. of Mathematics, University of Washington.Google Scholar