Advertisement

Extremes

, Volume 14, Issue 1, pp 127–152 | Cite as

Limit theorems for a recursive maximum process with location-dependent periodic intensity-parameter

Article
  • 60 Downloads

Abstract

We investigate the recursive sequence Z n : =  max {Z n − 1,λ(Z n − 1)X n } where X n is a sequence of iid random variables with exponential distributions and λ is a periodic positive bounded measurable function. We prove that the Césaro mean of the sequence λ(Z n ) converges toward the essential minimum of λ. Subsequently we apply this result and obtain a limit theorem for the distributions of the sequence Z n . The resulting limit is a Gumbel distribution.

Keywords

Césaro convergence Maximum process Random recursion Extremal value 

AMS 2000 Subject Classifications

60G70 60F05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Springer Series in Statistics. Springer, New York (1988)Google Scholar
  2. Denzel, G.E., O’Brien, G.L.: Limit theorems for extreme values of chain dependent processes. Ann. Probab. 3, 773–779 (1975)MathSciNetMATHCrossRefGoogle Scholar
  3. Doob, J.L.: Stochastic Processes. Wiley, New York (1953)MATHGoogle Scholar
  4. Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 288. Springer, Berlin (2003)MATHGoogle Scholar
  5. Lamperti, J.: On extreme order statistics. Ann. Math. Stat. 4, 1726–1737 (1964)MathSciNetCrossRefGoogle Scholar
  6. Resnick, S.I.: Extreme values, regular variation, and point processes. Appl. Probab. 4. Springer, New-York (1987)Google Scholar
  7. Resnick, S.I., Neuts, M.F.: Limit laws for maxima of a sequence of random variables defined on a Markov chain. Adv. Appl. Probab. 4, 285–295 (1970)CrossRefGoogle Scholar
  8. Spurny, K.R. (ed.): Advances in Aerosol Filtration. CRC, Boca Raton (1998)Google Scholar
  9. Turkman, K.F., Oliveira, M.F.: Limit laws for the maxima of chain dependent sequences with positive extremal index. J. Appl. Probab. 29, 222–227 (1992)MathSciNetMATHCrossRefGoogle Scholar
  10. Turkman, K.F., Walker, A.M.: Limit laws for the maxima of a class of quasi-stationary sequences. J. Appl. Probab. 20, 814–821 (1983)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany

Personalised recommendations