Let $\left\{\epsilon_{t}\right\}_{t \in \mathbb{Z}}$ be a sequence of iid random variables and define $X_{t}:=g(\ldots, \epsilon_{-1}, \epsilon_{0}, \epsilon_{1}, \ldots, \epsilon_{t})$. Let $\left\{\epsilon_{t}^{\prime}\right\}_{t \in \mathbb{Z}}$ be an iid copy of $\left\{\epsilon_{t}\right\}_{t \in \mathbb{Z}}$ and define $X_{t}^{\prime}:=g(\ldots, \epsilon_{-1}^{\prime}, \epsilon_{0}^{\prime}, \epsilon_{1}, \ldots, \epsilon_{t})$ be a coupled version of $X_{t}$. We say that $X_{t}$ is $\operatorname{GMC}(\alpha)$, $\alpha>0$, if there exist $C>0$ and $0<\rho=\rho(\alpha)<1$ such that, for all $t \in \mathbb{N}$, \begin{align*} \mathbb{E}\left[\left|X_{t}^{\prime}-X_{t}\right|^\alpha\right] \leq C \rho^t . \end{align*}
Suppose that I know that $X_t$ is GMC(2), stationary and mean-zero. How can I show that $|\gamma(k)|=O(\rho^k)$ where $\gamma(k)=\mathbb{E}[X_0X_k]$?
Thanks for your help!
It seems that we cannot hope for a better bound than $\lvert \gamma(k)\rvert=O(\rho^{k/2})$. Indeed, write $$ \gamma(k)=\mathbb E\left[X_0\left(X_k-X'_k\right)\right] +\mathbb E\left[X_0X'_k\right], $$ notice that $X_0$ is independent of $X'_k$ and use Cauchy-Schwarz inequality.
Taking $X_t=\sum_{k\geqslant 0}\rho^{-k/2}\varepsilon_{t-k}$ shows that this is the best we can get in general.