Suppose you have two instances of a distribution that are parameterized differently, and for one of them a certain restriction on the parameter values of the pdf or CDF results (perhaps after some algebraic manipulation) in a closed-form expression for the pdf or CDF (respectively) of another named distribution with one fewer parameters, as a special case.
Is it necessarily the case that the second parameterization also has a closed form of the same named distribution as a special case?
Take, for example, the scale and rate form of the gamma distribution. On one hand, these are merely alternative parameterizations of the same distribution, so it would seem that any distributions that nest in one must also nest in the other. On the other hand, I am not sure the limiting distributions are the same. Take the gamma distribution as an example. The rate parameterization and the scale parameterization seem to me to be different enough that distributions that are accessible only as limits might be different.
I am actually hoping that the answer is no: that the limiting distributions of a distribution are independent of parameterization.