The Infinitely Divisible Characteristic Function of Compound Poisson Distribution as the Sum of Variational Cauchy Distribution

Abstract

The new particular compound Poisson distribution is introduced as the sum of independent and identically random variables of variational Cauchy distribution with the number of random variables has Poisson distribution. This compound Poisson distribution is characterized by using characteristic function that is obtained by using Fourier-Stieltjes transform. The infinite divisibility of this characteristic function is constructed by introducing the specific function that satisfied the criteria of characteristic function. This characteristic function is employing the properties of continuity and quadratic form in term of real and nonnegative function such that its convolution has the characteristic function of compound Poisson distribution as the sum of variational Cauchy distribution.