Sufficient condition for a Poissonian stationary distribution

Theorem. The necessary and sufficient condition for a simple birth and death-type process to have a Poissonian stationary distribution (if it has a nondegenerate stationary distribution at all) is that it be a linear one (Erdi Toth, 1979). 

Sketch of the proof. If a simple birth and death-type process has a unique stationary distribution then its form is  

If the stochastic model of a chemical reaction can be identified with a simple birth and death process, and the process has stationary distribution, then it is precisely Poissonian, if the reaction is an open compartmental system. 

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