  "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
$\left(i,j\right)$ 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 $\left(i,j\right)$ 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.
