  For integer valued random variables, the translated Poisson distributions
form a flexible family for approximation in total variation, in much the same
way that the normal family is used for approximation in Kolmogorov distance.
Using the Stein--Chen method, approximation can often be achieved with error
bounds of the same order as those for the CLT. In this paper, an analogous
theory, again based on Stein's method, is developed in the multivariate
context. The approximating family consists of the equilibrium distributions of
a collection of Markov jump processes, whose analogues in one dimension are the
immigration--death processes with Poisson distributions as equilibria. The
method is illustrated by providing total variation error bounds for the
approximation of the equilibrium distribution of one Markov jump process by
that of another. In a companion paper, it is shown how to use the method for
discrete normal approximation in ${\mathbb Z}^d$.
