Strong approximation theorems for density dependent Markov chains

A variety of continuous parameter Markov chains arising in applied probability (e.g. epidemic and chemical reaction models) can be obtained as solutions of equations of the form where l[set membership, variant]Zt, the Y1 are independent Poisson processes, and N is a parameter with a natural interpretation (e.g. total population size or volume of a reacting solution). The corresponding deterministic model, satisfies X(t)=x0+ [integral operator]t0 [summation operator] lf1(X(s))ds Under very general conditions limN-->[infinity]XN(t)=X(t) a.s. The process XN(t) is compared to the diffusion processes given by and Under conditions satisfied by most of the applied probability models, it is shown that XN,ZN and V can be constructed on the same sample space in such a way that and