Almost sure stability of maxima

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.