  In 2010, Anderson, Craciun, and Kurtz showed that if a deterministically
modeled reaction network is complex balanced, then the associated stochastic
model admits a stationary distribution that is a product of Poissons
\cite{ACK2010}. That work spurred a number of followup analyses. In 2015,
Anderson, Craciun, Gopalkrishnan, and Wiuf considered a particular scaling
limit of the stationary distribution detailed in \cite{ACK2010}, and proved it
is a well known Lyapunov function \cite{ACGW2015}. In 2016, Cappelletti and
Wiuf showed the converse of the main result in \cite{ACK2010}: if a reaction
network with stochastic mass action kinetics admits a stationary distribution
that is a product of Poissons, then the deterministic model is complex balanced
\cite{CW2016}. In 2017, Anderson, Koyama, Cappelletti, and Kurtz showed that
the mass action models considered in \cite{ACK2010} are non-explosive (so the
stationary distribution characterizes the limiting behavior). In this paper, we
generalize each of the three followup results detailed above to the case when
the stochastic model has a particular form of non-mass action kinetics.
