The Distributions of the Mean of Random Vectors with Fixed Marginal Distribution

Abstract

Abstract Using recent results concerning non-uniqueness of the center of the mix for completely mixable probability distributions, we obtain the following result: For each $$d\in {\mathbb {N}}$$ d ∈ N and each non-empty bounded Borel set $$B\subset {\mathbb {R}}^d$$ B ⊂ R d , there exists a d -dimensional probability distribution $$\varvec{\mu }$$ μ satisfying the following: For each $$n\ge 3$$ n ≥ 3 and each probability distribution $$\varvec{\nu }$$ ν on B , there exist d -dimensional random vectors $${\textbf{X}}_{\varvec{\nu },1},{\textbf{X}}_{\varvec{\nu },2},\dots ,{\textbf{X}}_{\varvec{\nu },n}$$ X ν , 1 , X ν , 2 , ⋯ , X ν , n such that $$\frac{1}{n}({\textbf{X}}_{\varvec{\nu },1}+{\textbf{X}}_{\varvec{\nu },2}+\dots +{\textbf{X}}_{\varvec{\nu },n})\sim \varvec{\nu }$$ 1 n ( X ν , 1 + X ν , 2 + ⋯ + X ν , n ) ∼ ν and $${\textbf{X}}_{\varvec{\nu },i}\sim \varvec{\mu }$$ X ν , i ∼ μ for $$i=1,2,\dots ,n$$ i = 1 , 2 , ⋯ , n . We also show that the assumption regarding the boundedness of the set B cannot be completely omitted, but it can be substantially weakened.