Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-06-16T09:50:12.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic distributions of extremes of extremal Markov sequences

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

S. R. Adke  and
C. Chandran
S. R. Adke
Affiliation:
University of Poona
C. Chandran
Affiliation:
University of Poona
Get access Rights & Permissions [Opens in a new window]

Abstract

Let {ξn, n ≧1} be a sequence of independent real random variables, F denote the common distribution function of identically distributed random variables ξn, n ≧1 and let ξ1 have an arbitrary distribution. Define Xn+1 = k max(Xn, ξ n+1), Yn+ 1 = max(Yn, ξ n+1) – c, Un+1 = l min(Un, ξ n+1), Vn+1 = min(Vn, ξ n+1) + c, n ≧ 1, 0 < k < 1, l > 1, 0 < c < ∞, and X1 = Υ1= U1 = V1 = ξ1. We establish conditions under which the limit law of max(X1, · ··, Xn) coincides with that of max(ξ2, · ··, ξ n+1) when both are appropriately normed. A similar exercise is carried out for the extreme statistics max(Y1, · ··, Yn), min(U1,· ··, Un) and min(V1, · ··, Vn).

Keywords

DOMAIN OF ATTRACTIONMAXIMUM AND MINIMUM OF MARKOV SEQUENCES

MSC classification

Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Secondary: 60J05: Discrete-time Markov processes on general state spaces
Type
Short Communications
Information
Journal of Applied Probability , Volume 31 , Issue 1 , March 1994 , pp. 256 - 261
Copyright
Copyright © Applied Probability Trust 1994 

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.)

Footnotes

∗

Research supported by the National Board for Higher Mathematics, Bombay.

References

Alpuim, M. T. (1989) An extremal Markov sequence, J. Appl. Prob. 26, 219–232.Google Scholar
Chernick, M. R., Daley, D. J. and Littlejohn, R. P. (1988) A time reversibility relationship between two Markov chains with exponential stationary distributions, J. Appl. Prob. 25, 418–422.Google Scholar
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes, Academic Press, New York.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
Lewis, P. A. W. and Mckenzie, , Ed. (1991) Minification processes and their transformations. J. Appl. Prob. 28, 45–57.Google Scholar