The maximum term of uniformly mixing stationary processes

Let {X_n} be a uniformly (or strongly) mixing stationary process and let Z_n=max(X₁, X₂,..., X_n). For >0, let c_n()=inf {xR: n P(X₁>x)}. Under a condition which holds for all -mixing processes, necessary and sufficient conditions are given for P(Z_nc_n()) to converge to each possible limit. Some conditions for convergence of P(Z_nd_n) for any sequence d_n are also obtained.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.