On Upper and Lower Extremes in Stationary Sequences

Abstract

Unlike the classical result for iid sequences, the maximum and minimum of stationary mixing sequences can be asymptotically dependent. The class of non-degenerate limit distributions of the maximum and minimum from such processes turn out to be of the form H(x, ∞) — H(x, −y) where II(x, y) is a bivariate extreme value distribution. In some instances, the joint limiting distribution of any collection of upper and lower extremes may be determined as well. Moreover, using a special case of this result, it can be shown that the partial sums converge in distribution to a non-normal stable limit. Examples illustrating various aspects of the above results are also presented.