Large and Moderate Deviations for the Total Population Arising from a Sub-critical Galton–Watson Process with Immigration

Abstract

In this paper, we provide the exact forms of large and moderate deviations for the empirical mean of population and the centered total population of a sub-critical branching process with immigration. The rate functions in our large and moderate deviations are explicitly identified. Our theorems also apply to the models of the integer-valued autoregression. In computing the generating function requested by Gartner-Ellis theorem, our treatment substantially relies on an algorithm specifically designed for the autoregressive structure of our models.