Let $(X_{t})_{t \in \mathbb{N}}$ be a real-valued stationary stochastic process over probability $(\Omega,\mathcal{F},\mathbb{P})$, such that for $p\geq 2$, $X_{t} \in L_{p}(\mathbb{P})$ and it holds: \begin{equation} \label{eq:weak_dep} || E[X_{t+k}|\mathcal{F_{t}}] - E[X_{t+k}] ||_{p} \leq M_{p}\psi_{p}(k), \end{equation} where $\mathcal{F}_{t} = \sigma(X_{s}: s\leq t)$, $M_{p}:=||X_{t}-E[X_{t}]||_{p}$ , $\psi_{p}({\cdot})$ is a monotonically non-increasing function. Aforementioned condition is a mixingale like projective weak-dependency condition on $(X_{t})_{t\geq 1}$ (introduction in the book "Weak Dependence With Examples and Applications" by Dedecker et.al, more precisely Definition 2.7 on page 19 therein ). I would like to understand how to construct such a process if we are given a polynomial decay rate $\psi_{p}(k) = k^{-\gamma}$, $\gamma \in (0,\infty)$. Has anybody seen an example or could point me some relevant ideas?
