Convergence of moments of tau leaping schemes for unbounded Markov processes on integer lattices

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.