  Tau leap schemes were originally designed for the efficient time stepping of
discrete state and continuous in time Markov processes arising in stochastic
chemical kinetics. Previous convergence results on tau leaping schemes have
been restricted to systems that remain in a bounded subdomain (which may depend
on the initial condition) or satisfy global Lipschitz conditions on
propensities. This paper extends the convergence results to fairly general tau
leap schemes applied to unbounded systems that possess certain moment growth
bounds. Specifically, we prove a weak convergence result, which shows order $q$
convergence of all moments under certain form of moment growth bound
assumptions on the stochastic chemical system and the tau leap method, as well
as polynomial bound assumption on the propensity functions. The results are
stated for a general class of Markov processes with $\integ^N$ as their state
space.
