  We consider a type of nonnormal approximation of infinitely divisible
distributions that incorporates compound Poisson, Gamma, and normal
distributions. The approximation relies on achieving higher orders of cumulant
matching, to obtain higher rates of approximation error decay. The parameters
of the approximation are easy to fix. The computational complexity of random
sampling of the approximating distribution in many cases is of the same order
as normal approximation. Error bounds in terms of total variance distance are
derived. Both the univariate and the multivariate cases of the approximation
are considered.
