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Almost sure stability of maxima

Published online by Cambridge University Press:  14 July 2016

Sidney I. Resnick  and
R. J. Tomkins
Sidney I. Resnick*
Affiliation:
Technion — Israel Institute of Technology
R. J. Tomkins
Affiliation:
University of Saskatchewan, Regina
*
*Now at Stanford University.
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Abstract

For random variables {Xn, n ≧ 1} unbounded above set Mn = max {X1, X2, …, Xn}. When do normalizing constants bn exist such that Mn/bn→ 1 a.s.; i.e., when is {Mn} a.s. stable? If {Xn} is i.i.d. then {Mn} is a.s. stable iff for all and in this case bn ∼ F–1 (1 – 1/n) Necessary and sufficient conditions for lim supn→∞, Mn/bn = l > 1 a.s. are given and this is shown to be insufficient in general for lim infn→∞Mn/bn = 1 a.s. except when l = 1. When the Xn are r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.

Keywords

MAXIMA; ALMOST SURE STABILITYSEMI-MARKOV PROCESSESMARKOV CHAINS
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 2 , June 1973 , pp. 387 - 401
Copyright
Copyright © Applied Probability Trust 1973 

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