Poisson concentration proof of a density-dependent fluid limit ($Z^n(t)=Z^n(0)+{\sum }_j{v}_j{N}_j(n{\int }_0^t{\beta }_j(Z^n(u))du)/n$)

If the rate functions are bounded and Lipschitz on an invariant compact state space, the Poisson time-change representation of a Markov chain has drift $F={\sum }_j{v}_j{\beta }_j$. Each centered Poisson term is uniformly small after division by $n$. The Gronwall inequality then bounds the distance to $\dot{z}=F(z)$ by the initial error and the centered counting error, multiplied by $e^{LT}$. A Poisson maximal concentration bound gives exponentially decreasing error probabilities for a fixed horizon and fixed error tolerance.