On Laplace transforms of some probability densities (Q884156)

The author considers the probability density which consists of a normal product of the \(n\)-dimensional Cauchy density: \[ f(a,b,x)= c(a^2+ |x|^2)^{-(n+1)/2} (b^2+ |x|^2)^{-(n+ 1)/2}, \] where \(0< a< b\), \(n= 1,3,5,\dots\); \(x\in\mathbb{R}^n\) and \(c\) is a normalized constant. He also studies the density function \(g(3, u)\) defined by \[ {c\pi^{3/2}[u^{- 3/2}(\exp\{- a^2/u\}+ \exp\{- b^2/u\})- 2(b^2- a^2)^{-1} u^{-1/2}(\exp\{- a^2/u\}- \exp\{- b^2/u\})]\over (b^2- a^2)^2}. \] Assuming \(0< b^3- b^2 a- ba^2- a^3\), he proves that the probability distributions with density function \(g(3,u)\) and with density function \(f(a,b,x)\) are infinitely divisible.