  We prove estimates at infinity of convolutions $f^{n\star}$ and densities of
the corresponding compound Poisson measures for a class of radial decreasing
densities on $\mathbb{R}^d$, $d \geq 1$, which are not convolution equivalent.
Existing methods and tools are limited to the situation in which the
convolution $f^{2\star}(x)$ is comparable to initial density $f(x)$ at
infinity. We propose a new approach which allows one to break this barrier. We
focus on densities which are products of exponential functions and smaller
order terms -- they are common in applications. In the case when the smaller
order term is polynomial estimates are given in terms of the generalized Bessel
function. Our results can be seen as the first attempt to understand the
intricate asymptotic properties of the compound Poisson and more general
infinitely divisible measures constructed for such densities.
