The maximal correlation coefficient associated with the minimum

Abstract

For independent random variables $(X_i{)}_{1\le i\le n}$, we consider the maximal correlation coefficient $R=R({\min }_{i:1\le i\le m}{X}_i,{\min }_{j:\ell +1\le j\le n}{X}_j)$. If $X_1,X_2,\dots ,X_n$ are identically distributed with the same continuous distribution, we find that $R=(m-\ell )/\sqrt{m(n-\ell )}$. For discrete distributions, we calculate the maximal correlation coefficient $R$ for Bernoulli distributions, geometric distributions, binomial distributions and Poisson distributions. Our paper answers a question in \cite[Section~6]{ChangChen}.