  We derive new explicit bounds for the total variation distance between two
convolution products of $n$ probability distributions, one of which having
identical convolution factors. Approximations by finite signed measures of
arbitrary order are considered as well. We are interested in bounds with magic
factors, i.e. roughly speaking $n$ also appears in the denominator. Special
emphasis is given to the approximation by the $n$-fold convolution of the
arithmetic mean of the distributions under consideration. As an application, we
consider the multinomial approximation of the generalized multinomial
distribution. It turns out that here the order of some bounds given in Roos
(2001) and Loh (1992) can significantly be improved. In particular, it follows
that a dimension factor can be dropped. Moreover, better accuracy is achieved
in the context of symmetric distributions with finite support. In the course of
proof, we use a basic Banach algebra technique for measures on a measurable
Abelian group. Though this method was already used by Le Cam (1960), our
central arguments seem to be new. We also derive new smoothness bounds for
convolutions of probability distributions, which might be of independent
interest.
