Study Of Characterizations Of Infinitely Divisible Compound Poisson Distribution As The Sum Of Riemann Zeta Distribution

A huge body of deep results of infinite divisibility now exist in the literature.Infinitely divisible distributions are extensively used in applications,and they are also fundamentally related to the convergence of distributions of partial sums of independent random variables.Infinite divisibility of Riemann zeta distribution is studied by transforming its characteristic function to von Mangoldt function.Both Poisson distribution and Riemann zeta distribution are interesting examples of infinitely divisible distributions.Riemann zeta distribution can be combined with the Poisson distribution to form a compound Poisson distribution.Compound Poisson distribution is defined as the sum of independent and identically distributed random variables and a Poisson distribution.In this thesis,the compound Poisson distribution as the sum of independent and identically random variables from Riemann zeta distribution is characterized by using the characteristic function.This characteristic function is obtained by using Fourier-Stieljes transformation,and its infinite divisibility is constructed by introducing the specific function that satisfies the criteria of characteristic function.This thesis provides a characterization of compound Poisson distribution as the sum of Riemann zeta distribution and its infinite divisibility,and related characterizations are given,including the properties of characteristic function as continuous and positive definite function.