Large and moderate deviations for the total population arising from a sub-critical Galton-Watson process with immigration
Define a Galton-Watson process with immigration by the following relation \[ N_k = \sum_{j=1}^{N_k-1} \xi_{k-1,j} + \varepsilon_k, \] with i.i.d. offspring variables \(\xi_{k,j}\) and independent sequence of i.i.d. immigration variables \(\varepsilon_k\); with generic copies \(\xi\) and \(\varepsilon\), respectively. Assuming subcriticality \(E \xi < 1\) as well as finite exponential moments for \(\xi\) and \(\varepsilon\), the authors prove large and moderate deviation principles for empirical mean of the population size \[ \frac{N_1 + \dots + N_n}{n}, \] i.e., the empirical mean size of the generations \(1, \dots, n\).
scientific article; zbMATH DE number 6862617