Abstract
Consider K+(ν)={Q(m,ν)(dx);m∈(m0(ν),m+(ν))} the Cauchy-Stieltjes Kernel (CSK) family generated by a non degenerate probability measure ν with support bounded from above. Let X1,X2,…,Xn be boolean independent random variables with common distribution an element from K+(ν). i.e X1∼Q(m,ν) for some m∈(m0(ν),m+(ν)). We prove that the distribution of ∑i=1nXi belongs to the CSK family K+(ν) if and only if the probability measure ν is a scale transformation of the Marchenko–Pastur probability measure.