Hopefully this question is not too vague but I'm curious if there is some generalization of the stable and max-stable distributions when viewing them as limits of sums and maximums of random variables respectively. Essentially I'm wondering if you can define some conditions on a family of functions indexed by the naturals where after centering and scaling those functions applied to independent random variables (with reasonably general assumptions on the distribution) converges to some non-degenerate limit.
Its not even clear to me what the correct generalization is of summing or taking the maximum to even begin. My guess so far is something along the lines ofa family of functions $\{f_n\}_{n\geq 1}$ such that $f_n:\mathbb{R}^n\rightarrow\mathbb{R}$ and $f_n(x_1,\dots,x_n)=f_2(f_{n-1}(x_1,\dots,x_{n-1}),x_n)$.
This is just a curiosity, not a research or class related question so something even tangentially related would be really appreciated.