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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
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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 

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Footnotes

∗

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

References

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