I am currently taking the "Probability and Statistics" course and the asymptotic(limit) distribution appears in the context of Central Limit Theorem(CLT). Similar result also emerges from Fisher–Tippett–Gnedenko theorem. In the generalized form of the CLT, the requirement of finite moments is not necessary. Hence, by comparing CLT and Fisher–Tippett–Gnedenko theorem, I found the following heuristic relation:
$$
\text{i.i.d sample} \implies \text{existence of asymptotic(limit) distribution of certain class}
$$
Based on this relation, my question could be divided into two parts:
1. Is there any other cases where independent and identically-distributed sampling result in asymptotic(limit) distribution(s)?
2. How to explain the above relation in a more "fundamental" way? By saying "fundamental", I am expecting the answer can interpret the asymptotic(limit) distribution as a direct result from the nature of i.i.d sampling. It is understandable that higher-level tools from measure theory or functional analysis might be required to answer this question. I will do my best to comprehend those answers with the help of Wikipedia, so feel free to use any necessary mathematical tools.
[1]. https://en.wikipedia.org/wiki/Central_limit_theorem
[2]. https://en.wikipedia.org/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem