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We consider the approximation of a convolution of possibly differentprobability measures by (compound) Poisson distributions and also by relatedsigned measures of higher order. We present new total variation bounds having abetter structure than those from the literature. A numerical exampleillustrates the usefulness of the bounds, and an application in the Poissonprocess approximation is given. The proofs use arguments from Kerstan (Z.Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964) 173-179) and Roos (J.Multivariate Anal. 69 (1999) 120-134) in combination with new smoothnessinequalities, which could be of independent interest.

Refined total variation bounds in the multivariate and compound Poisson approximation