Extreme value theory for processes with periodic variances

Abstract

Suppose {Xn n ≥ 1} is a stationary sequence satisfying the D and D1 mixing conditions given by Adler [1]. Suppose further that g and h are functions on the positive integers such that h is positive, periodic with integer period p ≥ 1 and g satisfies a certain growth rate condition. By first exhibiting the weak convergence of a sequence of point processes related to , we derive the asymptotic distribution of . When {Xn} is in the maximal domain of attraction of A(x) = exp(-e -xx), {Mn} is attracted to (A(x))q/p where . If {Xn} is in the maximal domain of attraction of G(x) = exp(-x -α) then {Mn} is attracted to (G(x)) θ where . These asymptotic results lead to a new method for estimating the level of the k-year return period, a quantity of interest in many climatological studies. An analysis of the daily rainfall totals in Homestead, Florida from 1939-1979 is provided, illustrating this estimation procedure.