Extreme Value Theory for Suprema of Random Variables with Regularly Varying Tail Probabilities

摘要

Extreme value theory concerns the joint tail behavior and related problems of random variables (r.v.'s). Recent emphasis has been the extension of the classical theory, which considers independent and identically distributed (i.i.d.) r.v.'s to the more general setting of stationarity. Progress has been made on topics such as notions of asymptotic independence, general extremal types theorems, studies of related point processes, etc. The author is interested in the extremal properties of stationary sequences whose members are certain functions of i.i.d. r.v.'s. In this direction, previous documents investigated moving average sequences under various assumptions. Through the particular structure of the sequences, these studies provided invaluable insights into the theory in general. This paper considers a stationary sequences (X sub j) consisting of the seighted suprema -- instead of sums as in the case of moving average -- of certain i.i.d. r.v.'s whose tail probabilities are regularly varying. A sequence with this structure may be used to model random exchanges and is a useful tool in studying multivariate extreme value theory.