Suppose the process $\{X_{t}:t\in Z\}$ is absolutely continuous distributed and strong mixing with the coefficient $\alpha(s)$ defined as \begin{equation} \alpha(s) \equiv \sup \{ |P(A\cap B) - P(A)P(B)| : -\infty < t<\infty, A \in X^{t}_{-\infty}, B\in X^{\infty}_{t+s} \}. \end{equation} Here $\alpha(s) \rightarrow 0$ at a polynomial rate.
Now, consider a new process of infinitely distributed lags as $Y_{t} = \lambda(L)X_{t} \equiv X_{t} + \lambda_{1}X_{t-1} + \lambda_{2} X_{t-2} + \ldots$, where $\lambda_{k} = \rho^{k}$ with $|\rho|<1$.
My question is: is the process $Y_{t}$ strong mixing (or under what conditions)? If $Y_{t}$ can be strong mixing, what is the mixing coefficient of $Y_{t}$? The same as $\alpha(s)$？
In general no. In the paper
Donald W. K. Andrews. “Non-Strong Mixing Autoregressive Processes.” Journal of Applied Probability, vol. 21, no. 4, 1984, pp. 930–934. JSTOR, www.jstor.org/stable/3213710.,
it is shown that even in the case where $\left(X_t\right)_{t\in\mathbb Z}$ is i.i.d. with a Bernoulli marginal distribution, the process $\left(Y_t\right)_{t\in\mathbb Z}$ is not mixing in general.
There exists conditions in the absolutely continuous case, which are contained in the paper
Withers, C.S. Z. Wahrscheinlichkeitstheorie verw Gebiete (1981) 57: 477. https://doi.org/10.1007/BF01025869