Optimal CLT error bounds in the second order Wasserstein metric via Stein's method for weakly dependent sequences

We use Stein’s method to prove general upper bounds on the second order Wasserstein distance in the CLT for general dependent partial sums $S_N=X_1+\dots +X_N$. Then we use these estimates to provide optimal $O(N^{-1\mathrm{/}2})$ rates for a variety of chaotic dynamical systems and $\varphi$- and $\psi$-mixing sequences. Our results for $\psi$-mixing sequences are obtained under the fourth moment condition, which matches the best known results in the iid case. This is the first time that optimal CLT rates of any kind are obtained for unbounded weakly stationary variables under general mixing conditions. We then apply this result and obtain optimal CLT rates $O((\text{Var}(S_N){)}^{-1\mathrm{/}2})$ for non-stationary $\psi$-mixing sequences without any growth rates on $\text{Var}(S_N)$. We also obtain close to optimal CLT rates for nonconventional sums and some classes of random dynamical systems with non-uniform decay of correlations