On Eigenvalues of the Transition Matrix of some Count Data Markov Chains

A stationary Markov chain is uniquely determined by its transition matrix, the eigenvalues of which play an important role for characterizing the stochastic properties of a Markov chain. Here, we consider the case where the monitored observations are counts, i.e., having values in either the full set of non-negative integers, or in a finite set of the form ${0,...,n}$ with a prespecified upper bound $n$. Examples of count data time series as well as a brief survey of some basic count data time series models is provided.

Then we analyze the eigenstructure of count data Markov chains. Our main focus is on so-called CLAR(1) models, which are characterized by having a linear conditional mean, and also on the case of a finite range, where the second largest eigenvalue determines the speed of convergence of the forecasting distributions. We derive a lower bound for the second largest eigenvalue, which often (but not always) even equals this eigenvalue. This becomes clear by deriving the complete set of eigenvalues for several specific cases of CLAR(1) models. Our method relies on the computation of appropriate conditional (factorial) moments.