The maximal correlation coefficient associated with the minimum

TL;DR

The paper determines the maximal correlation between the minima of two blocks of independent variables. It proves a sharp bound $R=(m-ℓ)/\sqrt{m(n-ℓ)}$ for continuous i.i.d. distributions and derives explicit formulas for discrete cases: Bernoulli, geometric, binomial, and Poisson, with Poisson obtained as a limit of binomial. The approach combines foundational properties of maximal correlation, a monotone-transform technique to the exponential case, and discrete reductions to Bernoulli correlations (via Csaki-Fischer). The results resolve questions from ChangChen and provide exact dependence measures for these common distributions, enriching the understanding of order-statistic–based correlations. Practical impact lies in precise dependence quantification for minima of independent samples across several distribution families.

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}.

Theorems & Definitions (13)

• Proposition 1.1: Proposition 5.5 in ChangChen
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• Remark 3.1
• Lemma 4.1
• Remark 4.1