Linear transformations that preserve majorization, Schur concavity, and exchangeability
A vector \(w\in R^ n\) is said to majorize another vector \(x\in R^ n\) (notation: \(w>x)\) if \(\sum^{n}_{i=1}w_ i=\sum^{n}_{i-1}x_ i\) and \(\sum^{k}_{i=1}w_{[i]}\geq \sum^{k}_{i=1}x_{[i]}\) for each \(k=1,2,...,n-1\) where \(x_{[1]}\geq x_{[2]}\geq...\geq x_{[n]}\) denote the components of x in decreasing order. A random vector \(x\in R^ n\) is said to be exchangeable if R(\(\tau\))x has the same distribution as x for any permutation matrix R(\(\tau\)). A linear transformation A: \(R^ m\to R^ n\) is said to preserve majorization if \(Ay>Az\) whenever \(y>z.\) Let \(m\leq n\), A: \(R^ m\to R^ n\), B: \(R^ n\to R^ m\), A and B have rank m. Condition 1: for every permutation \(\pi\) there exists a permutation \(\tau\) such that \(R(\tau)A=AR(\pi).\) Condition 2: for every permutation \(\pi\) there exists a permutation \(\sigma\) such that \(BR(\sigma)=R(\pi)B.\) First the authors find the explicit forms of matrices that satisfy conditions 1 and 2. Then they prove several theorems that relate conditions 1 and 2 to majorization, exchangeability and Schur concavity, for example: the transformation A preserves majorization iff it satisfies condition 1 (Theorem 3); the transformation B preserves exchangeability iff it satisfies condition 2. Some of the results are generalized for the case when the group of permutations is replaced by its subgroup. Using the algebraic results the authors present some applications to multivariate density functions.
scientific article; zbMATH DE number 4171108