The evolution of a batch-immigration death process subject to counts

A bivariate batch immigration-death process is developed to study the degree to which the fundamental structure of a hidden stochastic process can be inferred purely from counts of escaping individuals. This question is of immense importance in fields such as quantum optics, where externally based radiation elucidates the nature of the underlying electromagnetic radiation process. Batches of i immigrants enter the population at rate αqi, and each individual dies independently at rate μ. General expressions are developed for the population size cumulants and probabilities, together with those for the associated counting process. The strong link between these two structures is highlighted through two specific examples, involving k-batch immigration for i=k, and Schoenberg-batch immigration over i=2m (m =0, 1, 2, ...), and shows that high quality inferences on the hidden population process can be inferred purely from externally counted observations.