  It has been known for nearly a decade that deterministically modeled reaction
networks that are weakly reversible and consist of a single linkage class have
trajectories that are bounded from both above and below by positive constants
(so long as the initial condition has strictly positive components). It is
conjectured that the stochastically modeled analogs of these systems are
positive recurrent. We prove this conjecture in the affirmative under the
following additional assumptions: (i) the system is binary and (ii) for each
species, there is a complex (vertex in the associated reaction diagram) that is
a multiple of that species. To show this result, a new proof technique is
developed in which we study the recurrence properties of the $n-$step embedded
discrete time Markov chain.
