There are a lot of central limit theorems (CLT) that can be applied to weak dependent arrays. Unfortunately, I am working on a particular array and according to my knowledge, there is no CLT I can use.
Let $X_{i,n}$ be a process of iid random real-valued variables. Let $Y_{i,n}$ be another process, such that $Y_{i,n} = |X_{i,n}|^{1+\epsilon/n^{0.4}}$, where $\epsilon$ is a random variable assumed $O_p(1)$. I need the asymptotic distribution of $n^{-1/2}\left(S_n - E(S_n)\right)$, where $S_n = \sum_{i=1}^n Y_{i,n}$.
My conjecture is that the limit distribution would be $N(0, \Sigma)$, where $\Sigma$ is the variance limit of the variance of $n^{-1/2}\left(S_n\right)$, as $n$ grows to infinity. I used simulations with several examples that suggest this result. For instance, $\epsilon$ follows Poisson distribution, Normal distribution, etc.
Is there a CLT that can be applied to arrays of random variables that are independent only when $n = \infty$?