Nonnormal small jump approximation of infinitely divisible distributions

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.