Certain bivariate distributions and random processes connected with maxima and minima

Abstract

The minimum and the maximum of t independent, identically distributed random variables have (F) over bar (t) and F-t for their survival (minimum) and the distribution (maximum) functions, where (F) over bar = 1 - F and F are their common survival and distribution functions, respectively. We provide stochastic interpretation for these survival and distribution functions for the case when t > 0 is no longer an integer. A new bivariate model with these margins involve maxima and minima with a random number of terms. Our construction leads to a bivariate max-min process with t as its time argument. The second coordinate of the process resembles the well-known extremal process and shares with it the one-dimensional distribution given by F-t. However, it is shown that the two processes are different. Some fundamental properties of the max-min process are presented, including a distributional Markovian characterization of its jumps and their locations.