I'm wondering if there is a large deviations result for asymptotically independent and identical distributed (a.i.i.d.) variables.
If $\{(X^i_n)_{i=1,\dots,n}\}_{n\in\mathbb{N}}$ is a sequence of a.i.i.d. variables, does a CLT/large deviation principle exist for the sum of $S_n=\sum_{i=1}^n X^i_n$?
For a.i.i.d. let's put it in this way (taken more or less from here [1, chapter 6.3]): Fix $m\in\mathbb{N}$ and suppose that, for each $n\in\mathbb{N}$, you have a random vector $(D^{(n)}_i)_{i\in[m]}$, where $[m]=\{1,\dots ,m \}$. We say that $D^{(n)}_i$ are asymptotically independent if we can couple the vector $(D^{(n)}_i)_{i\in[m]}$ to an independent vector $(\hat{D}^{(n)}_i)_{i\in[m]}$ such that
$\lim_{n\rightarrow \infty} \mathbb{P} \left((D^{(n)}_i)_{i\in[m]} \neq (\hat{D}^{(n)}_i)_{i\in[m]} \right)=0$.
Do you think I can extended this definition to $m=n$ and get a definition of a.i.i.d. if the $\hat{D}^{(n)}_i$ are i.i.d. for all $n$?