Poisson-process limit laws yield Gumbel max-min and min-max

Abstract

A chain is only as strong as its weakest link" says the proverb. But what about a collection of statistically identical chains: How long till all chains fail? The answer to this question is given by the Max-Min of a matrix whose (i, j) entry is the failure time of link j of chain i: take the minimum of each row, and then the maximum of the rows' minima. The corresponding Min-Max is obtained by taking the maximum of each column, and then the minimum of the columns' maxima. The Min-Max applies to the storage of critical data. Indeed, consider multiple backup copies of a set of critical data items, and consider the (i, j) matrix entry to be the time at which item j on copy i is lost; then, the Min-Max is the time at which the first critical data item is lost. In this paper we address random matrices whose entries are independent and identically distributed random variables. We establish Poisson-process limit-laws for the row's minima and for the columns' maxima. Then, we further establish Gumbel limit-laws for the Max-Min and for the Min-Max. The limit-laws hold whenever the entries' distribution has a density, and the Gumbel limit-laws yield highly applicable approximation tools and design tools for large random matrices.