Let $(X_i)_{i \in \mathbb{Z}}$ be i.i.d. with $X_1 \sim Be(p), p \in [0,1]$. Define $Y_i^{(p)} = 1_{\{X_{i-1}=X_i=1\}}, i \in \mathbb{Z}$.
I would like to construct a coupling $\left(\widehat{Y^{(p)}_i}, \widehat{Y^{(p^{\prime})}_i} \right)$ of $Y_i^{(p)}$ and $Y_i^{(p^{\prime})}$such that $P\left(\widehat{Y^{(p)}_i} \leq \widehat{Y^{(p^{\prime})}_i}\right) = 1$ provided that $p \leq p^{\prime}$.
The definition of a coupling of $n$ random variables $X_i : (\Omega_i, \mathcal{A}_i, P_i) \to (E_i, \mathcal{E}_i)$ is a vector $\hat{X} = (\hat{X}_1, \dotsc, \hat{X}_n)$ where $$\hat{X} : (\Omega, \mathcal{A}, P) \to \left(\bigotimes_{i=1}^nE_i, \bigotimes_{i=1}^n\mathcal{E}_i\right),$$ such that the law $\mu_{\hat{X}_i} = \mu_{X_i}$ for any $i = 1, \dotsc, n$.
So I thought maybe I should determine the law(distribution) of $Y_i^{(p)}$ first. Basically, $Y_i^{(p)}$ is again an indicator function right? So, $$Y_i^{(p)} = \begin{cases} 1 \ \ \ \ \text{if} \ y \in {\{X_{i-1} = 1\}\cap\{X_i=1\}}\\ 0 \ \ \ \ \text{if} \ y \in (\{X_{i-1} = 1\}\cap\{X_i = 0\})\ \cup \ (\{X_{i-1} = 0\}\cap\{X_i = 1\}) \end{cases} $$ Is this information useful for constructing a coupling? Is this even correct? If not, anyone has an idea/hint how I can construct such a coupling?