Asymptotic Independent Representations for Sums and Order Statistics of Stationary Sequences

Preface This is a report on the author's research initiated during the stay in GG ottingen in 1987/88 and continued over the past few years. The question posed at the very beginning was simple: to identify the area where we can still apply the classical limit theory for sums and maxima of independent random variables. Existence of asymptotic independent representations became a natural frame for the considerations. Obtained this way theory is a uniication of several trends in limit theory for sums and helps in understanding the asymptotic structure of order statistics. This is not a monograph on limit theory for sums and order statistics of dependent stationary random variables. The choice of presented results and examples depends heavily on the author's taste and interest. In particular, we concentrate on results which we can obtain by means developed in the paper, and on examples demonstrating how weak are the assumptions usually made in the paper. The author wishes to express his gratitude to the Alexander von Humboldt Foundation for supporting his stay in GG ottingen and for active help every year since that time. iii iv CONTENTS II Order statistics 65 6 Asymptotic independent representations for maxima 67 6.1 Main