On extreme values in stationary sequences

Abstract

In this paper, extreme value theory is considered for stationary sequences ζn satisfying dependence restrictions significantly weaker than strong mixing. The aims of the paper are: (i) To prove the basic theorem of Gnedenko concerning the existence of three possible non-degenerate asymptotic forms for the distribution of the maximum Mn = max(ξ1... ξn), for such sequences. (ii) To obtain limiting laws of the form {Mathematical expression} where Mn(r)is the r-th largest of ξ1... ξn, and Pr ξ1>un∼Τ/n. Poisson properties (akin to those known for the upcrossings of a high level by a stationary normal process) are developed and used to obtain these results. (iii) As a consequence of (ii), to show that the asymptotic distribution of Mn(r)(normalized) is the same as if the {ξn} were i.i.d. (iv) To show that the assumptions used are satisfied, in particular by stationary normal sequences, under mild covariance conditions. © 1974 Springer-Verlag.