The Infinitely Divisible Compound Negative Binomial Distribution as the Sum of Laplace Distribution

The infinite divisibility of compound negative binomial distribution, especially as the sum of Laplace distribution has important roles in governing the mathematical model based on its characteristic function. In order to show the property of characteristic function of this compound negative binomial distribution, it is used Fourier-Stieltjes transform to have characteristic function. The characteristic function property is governed to show the continuity and quadratic form by using analytical approaches. The infinite divisibility property is obtained by introducing a function satisfied the criteria to be a characteristic function such that its convolution has the characteristic function of compound negative binomial distribution. Then it is concluded that the characteristic function of compound negative binomial distribution as the sum of Laplace distribution satisfies the property of continuity, quadratic form and infinite divisibility.