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

Exponential densities and compound Poisson measures