Abstract
Let {X n } be a uniformly (or strongly) mixing stationary process and let Z n =max(X 1, X 2,..., X n ). For ξ>0, let c n (ξ)=inf {xεR: n P(X 1>x)≦ξ}. Under a condition which holds for all ϕ-mixing processes, necessary and sufficient conditions are given for P(Zn≦cn(ξ)) to converge to each possible limit. Some conditions for convergence of P(Zn≦dn) for any sequence d n are also obtained.
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Research supported in part by the National Research Council of Canada and done at the Summer Research Institute of the Canadian Mathematical Congress.
We are grateful to Professor D.L. McLeish and the referee for some useful comments, particularly in connection with Lemma 2.
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O'Brien, G.L. The maximum term of uniformly mixing stationary processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 30, 57–63 (1974). https://doi.org/10.1007/BF00532863
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DOI: https://doi.org/10.1007/BF00532863