  We introduce a class of two-parameter discrete dispersion models, obtained by
combining convolution with a factorial tilting operation, similar to
exponential dispersion models which combine convolution and exponential
tilting. The equidispersed Poisson model has a special place in this approach,
whereas several overdispersed discrete distributions, such as the Neyman Type
A, P\'olya-Aeppli, negative binomial and Poisson-inverse Gaussian, turn out to
be Poisson-Tweedie factorial dispersion models with power dispersion functions,
analogous to ordinary Tweedie exponential dispersion models with power variance
functions. Using the factorial cumulant generating function as tool, we
introduce a dilation operation as a discrete analogue of scaling, generalizing
binomial thinning. The Poisson-Tweedie factorial dispersion models are closed
under dilation, which in turn leads to a Poisson-Tweedie asymptotic framework
where Poisson-Tweedie models appear as dilation limits. This unifies many
discrete convergence results and leads to Poisson and Hermite convergence
results, similar to the law of large numbers and the central limit theorem,
respectively. The dilation operator also leads to a duality transformation
which in some cases transforms overdispersion into underdispersion and
vice-versa. Many of the results have multivariate analogues, and in particular
we consider a class of multivariate Poisson-Tweedie models, a multivariate
notion of over- and underdispersion, and a multivariate zero-inflation index.
