The maximal correlation coefficient associated with the minimum

TL;DR

This paper calculates the maximal correlation coefficient between the minimums of independent random variables, providing explicit formulas for continuous and discrete distributions, thus answering a specific open question.

Contribution

It derives explicit formulas for the maximal correlation coefficient between minima of independent variables, including continuous and discrete cases, addressing an open problem.

Findings

For i.i.d. continuous variables, R = (m - ℓ)/√(m(n - ℓ))

Explicit R values for Bernoulli, geometric, binomial, and Poisson distributions

Provides solutions to an open question in the literature

Abstract

For independent random variables $(X_i)_{1\leq i\leq n}$, we consider the maximal correlation coefficient $R=R(\min_{i:1\leq i\leq m}X_i,\min_{j:\ell+1\leq j\leq n}X_j)$. If $X_1,X_2,\ldots,X_n$ 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}.