  We consider the approximation of a convolution of possibly different
probability measures by (compound) Poisson distributions and also by related
signed measures of higher order. We present new total variation bounds having a
better structure than those from the literature. A numerical example
illustrates the usefulness of the bounds, and an application in the Poisson
process 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 smoothness
inequalities, which could be of independent interest.
