A decomposition of the beta distribution, related order and asymptotic behavior
Let \(\beta_{v,w}\) be a random variable distributed according to the beta distribution with parameters v and w. The authors prove that \(U_{v,w}=^{def}-\log \beta_{v,w}=U^{CM}+U^{PF}\), where \(U^{CM}\) and \(U^{PF}\) are independent with completely monotone and \(PF_{\infty}\) densities, respectively. It is shown that \(U_{v,w}\) is infinitely divisible and \(\beta_{v,w}\) correspondingly infinitely factorizable. The asymptotic behavior of \(U_{v,w}\) and \(\beta_{v,w}\) for different modes of increase of v and w is studied. The decomposition is employed to provide an algorithm for generating random beta distributed numbers.
scientific article; zbMATH DE number 3842865
scientific article