On eigenvalues of the transition matrix of some count-data Markov chains (Q1707062)

This paper studies the spectrum of the transition matrix of CLAR(1) models. The homogeneous Markov chain \((X_t)_{\mathbb N_0}\) is said to have CLAR(1) structure if the conditional mean of \(X_t\) is linear in \(X_{t-1}\). It is shown that the corresponding linear coefficient is an eigenvalue of the transition matrix. This result yields a lower bound for the modulus of second largest eigenvalue, which determines the rate of convergence of the forecasting distributions. Some specific cases of CLAR(1) models for count-data Markov chains are further investigated here. By computing the appropriate conditional moments, the complete set of eigenvalues of the transition matrix is derived for the models INAR(1), Poisson INARCH(1), binomial AR(1), binomial INARCH(1), and beta-binomial AR(1).