2010/09/24 by Nizar Demni, Demni, Nizar
Economics, Econometrics and Finance · Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Random Matrices and Applications #Statistics Theory (math.ST) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1009.4926
openalex publication_date 2010/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We derive the representative Bernstein measure of the density of (Xα)-α/(1-α), 0 < α< 1, where Xα is a positive stable random variable, as a Fox-H function. When 1-α= 1/j for some integer j ≥ 2, the Fox H-function reduces to a Meijer G-function so that the Kanter's random variable (see below) is closely related to a product of (j-1) independent Beta random variables. When α tends to 0, the Bernstein measure becomes degenerate thereby agrees with Cressie's result for the asymptotic behaviour of stable distributions for small values of α. Coming to free probability, our result makes more explicit that of Biane on the density of its free analog. The paper is closed with analytic arguments explaining the occurence of the Kanter's random variable in both the classical and the free settings.