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Laguerre population processes

We analyse a number of stochastic processes that give rise to first-order number statistics governed by Laguerre distributions with properties that lie between Poisson and geometric random variables. These distributions have hitherto been used to characterize the photon statistics of a coherent mixture of thermal and laser light. Here, we explore a number of discrete population processes that can lead to the same first-order statistics and correlation functions and highlight the distinguishing features that may be used to identify the relevant model from data.