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An extremal markovian sequence

Published online by Cambridge University Press:  14 July 2016

M. Teresa Alpuim
M. Teresa Alpuim*
Affiliation:
University of Lisbon and CEA (INIC)
*
∗ Postal address: DEIOC, University of Lisbon, 58 Rua da Escola Politénica, 1294 Lisboa Codex, Portugal.
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Abstract

In this paper we consider an independent and identically distributed sequence {Yn} with common distribution function F(x) and a random variable X0, independent of the Yi's, and define a Markovian sequence {Xn} as Xi = X0, if i = 0, Xi = k max{Xi− 1, Yi}, if i ≧ 1, k ∈ R, 0 < k < 1. For this sequence we evaluate basic distributional formulas and give conditions on F(x) for the sequence to possess a stationary distribution. We prove that for any distribution function H(x) with left endpoint greater than or equal to zero for which log H(ex) is concave it is possible to construct such a stationary sequence with marginal distributions equal to it. We study the limit laws for extremes and kth order statistics.

Keywords

STATIONARY DISTRIBUTIONLIMIT LAWSEXTREME VALUESORDER STATISTICS
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 2 , June 1989 , pp. 219 - 232
Copyright
Copyright © Applied Probability Trust 1989 

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References

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