  Count data are omnipresent in many applied fields, often with overdispersion.
With mixtures of Poisson distributions representing an elegant and appealing
modelling strategy, we focus here on how the tail behaviour of the mixing
distribution is related to the tail of the resulting Poisson mixture. We define
five sets of mixing distributions and we identify for each case whenever the
Poisson mixture is in, close to or far from a domain of attraction of maxima.
We also characterize how the Poisson mixture behaves similarly to a standard
Poisson distribution when the mixing distribution has a finite support.
Finally, we study, both analytically and numerically, how goodness-of-fit can
be assessed with the inspection of tail behaviour.
