Thanks Luca! Indeed the gamma distribution is a good fit to my simulations. Did you have any sources or papers, where I can find more information on this?

Nevertheless, thank you for your help, Luca! Gamma-distribution is a good approximation for compounding Maxwellian distribution, where the scale parameter is (uniformly) distributed in the range of at least +-1.5 times the mean scale parameter. The tails differ at very low probabilities, however. But that will do for my purposes. For a very small range of scaling parameters the Maxwellian distribution itself is a sufficient approximation as a compound. So your hint to gamma, although the confusion, was quite helpful.