Just a reference on which all $\rho_i$ are defined on. e.g. $D_i = [-i, i], i > 1$ and $D = [-1,1]$ or something. I often see this in literature where you might have some normal looking distribution on an interval $[-x, x]$ with non-zero centre but then the distribution is shifted to lie on say $[-1,1]$. But I don't see how exactly this is done in general since you would be required to renormalise (I guess it doesn't matter?).

The other part of the question is about how I would go about finding a transformation which relates the family of densities obtain one density from the other: $\rho_{i+1}(n) = f( \rho_i(n))$. Any standard examples?