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

Multivariate approximation in total variation, I: equilibrium distributions of Markov jump processes