Characterization problems for linear forms with free summands

Abstract

Let $T_1,...,T_n$ denote free random variables. For two linear forms $L_1={\sum }_{j=1}^n{a}_j{T}_j$ and $L_2={\sum }_{j=1}^n{b}_j{T}_j$ with real coefficients $a_j$ and $b_j$ we shall describe all distributions of $T_1,...,T_n$ such that $L_1$ and $L_2$ are free. For identically distributed free random variables $T_1,...,T_n$ with distribution $\mu$ we establish necessary and sufficient conditions on the coefficients $a_j,b_j,j=1,...,n,$ such that the statements:\quad $(i)$ $\mu$ is a centered semicircular distribution; and $(ii)$ \, $L_1$ and $L_2$ are identically distributed ($L_1\stackrel{D}{=}{L}_2$); are equivalent. In the proof we give a complete characterization of all sequences of free cumulants of measures with compact support and with a finite number of non zero entries. The characterization is based on topological properties of regions defined by means of the Voiculescu transform $\varphi$ of such sequences.

References in corpus (3)

• Cumulants in noncommutative probability I. Noncommutative Exchangeability Systems
• Cumulants in noncommutative probability II. Generalized Gaussian random variables
• On superconvergence of sums of free random variables

Cited by in corpus (4)

• Cauchy-Stieltjes families with polynomial variance functions and generalized orthogonality
• Sample Variance in Free Probability
• Free Component Analysis: Theory, Algorithms & Applications
• Probability distributions with rational free $R$-transform