Suppose I have a random sequence $\mathbf{X}=\{X_n\}_{n\in\mathbb{Z}}\subset \mathbb{R}^{\mathbb{Z}}$ that is ergodic with respect to translations. I am interested in a sufficient condition on $\mathbf{X}$ that would imply that $\sum_{n=1}^NX_n \sim \mu N + \eta_N N^{\frac{1}{2}}$ where $\mathbb{E}(X_1)=\mu$ and $\eta_N$ is a sequence of tight random variables such that $\eta_N \nrightarrow \delta_0$. Mixing/Markov/Martingale could be hard to come by. Are there other sufficient conditions for square root fluctuations? does giving up $\eta_N \rightarrow \mathcal{N}(0,\sigma^2)$ help in any way?
