I have a stochastic process $\{W_t\}_{t\geq 1}$, of uncorrelated but not indipendent random variables, with $\mathbb{E}(W_t) = 0$ and $Var(W_t)=\frac{t-1}{2}$ $\forall, t\geq 1$ (The $\{W_t\}_{t\geq 1}$ are not identically distributed). In particular I have some evidence (by Monte Carlo simulation) that: $$ D_n = 1+\frac{2}{n}\sum_{t = 1}^{n}W_t\to Z\sim Exp(1) \mbox{ as $n\to\infty$} $$ where $"\to"$ represents the convergence in distribution of the new process $D_n$. By simply calculations $\mathbb{E}(D_n) = 1$ and $Var(D_n) = 1-\frac{1}{n}$. Classic limit theorems do not apply in this case, because $W_t$ are not indipendent random variables (although uncorrelated).
I have just read about the $\alpha$-mixing condition of a stochastic process, and in general the concept of "measure of indipendence" and "asymptotic indipendence". You can find the main definitions for example here:
https://encyclopediaofmath.org/wiki/Strong_mixing_conditions
I have seen that there are many limit theorems which use this property to prove that some kind of sequence converge in distribution and I really think that one of them will be the key to my case. For example in the following book:
there are some of them.
If needed I will add the definition of $W_t$, although I would prefer a general theorem which also covers my particular case (with the assumption that $W_t$ or $D_n$ satisfy the $\alpha$-mixing condition).
Question
Do you know a limit theorem which use (or doesn't use) the $\alpha$-mixing condition and such that my case satisfies its hypothesis? Thank you in advance for your help!