Peculiarities of finding characteristic functions of the generating process in the model of stationary linear AR(2) process with negative binomial distribution

The linear random AR(2) autoregressive process having the negative binomial distribution has been considered. It has the form ξ _t + a ₁ξ _t-1 + a ₂ξ _t-2 = ? _t , t ∈ Z, where {a ₁, a ₂ ≠ 0} are the autoregressive parameters; Z = {...,-1,0,1,...} is the sequence of integers; {ξ _t ,t ∈ Z} is the random process with discrete time and independent values having the infinitely divisible distribution law that is called generating process. The method of finding the characteristic function of the generating process for linear autoregressive process having negative binomial distribution is presented. This inverse problem is solved by using properties of the characteristic function of stationary linear autoregressive process that can be presented in the Kolmogorov canonical form and as a linear stationary autoregressive process. An example of finding the Poisson spectrum of jumps and the characteristic function for the linear second order autoregressive process (AR(2)) with negative binomial distribution has been also presented.