Almost sure convergence of the maximum of a stationary sequence and asymptotic properties of probability weighted moments.

摘要

This thesis is divided into two parts. The first part is devoted to the study of the almost sure convergence of the maximum of a stationary sequence. The second part addresses some theoretical issues concerning probability weighted moments.;The almost sure convergence of the maximum has been mostly studied in the independent case. Resnick and Tomkins (1973) showed that the maximum of a sequence of independent and identically distributed random variables with common distribution function F behaves asymptotically like the quantile F-1(1 - 1 / n) whenever the distribution decreases quickly enough. In this thesis, the independence assumption is removed and instead, we suppose that the sequence is stationary. Then a natural question is to ask if the maximum still behaves asymptotically like the quantile F-1(1 - 1/n). This is not true in general.;The principal goal of the first part of this thesis is to determine sufficient and necessary conditions for the almost sure convergence of the maximum of a stationary sequence. We will show that if a particular mixing condition is satisfied then the maximum behaves asymptotically like the quantile F-1(1 - 1 /n ). More precisely, the required mixing condition will allow us to break down the stationary sequence into asymptotically independent blocks. The method of construction of each block is borrowed from Klass (1984) who used a similar technique for the independent case. Applications relative to Markov chains and normal stationary sequences will be presented.;In the second part of the thesis, we will first show that the factorial moment bound is better than Chernoff's bound for discrete non-negative random variables. Sharper bounds which are function of probability weighted moments (introduced by Greenwood et al. (1979)) are also studied extensively. An equivalence of the upper tail of the distribution function of a non-negative random variable with finite moments will be obtained in terms of probability weighted moments. The proof for getting this equivalence is based on a Tauberian type argument. Relationships between stochastic orders and probability weighted moments will be also studied. To illustrate our findings, applications for dynamic programming and reliability theory will be presented. and reliability theory will be presented.