This paper focuses on estimating the invariant density function $f_X$ of the strongly mixing stationary process $X_t$ in the multiplicative measurement errors model $Y_t = X_t U_t$, where $U_t$ is also a strongly mixing stationary process. We propose a novel approach to handle non-independent data, typical in real-world scenarios. For instance, data collected from various groups may exhibit interdependencies within each group, resembling data generated from $m$-dependent stationary processes, a subset of stationary processes. This study extends the applicability of the model $Y_t = X_t U_t$ to diverse scientific domains dealing with complex dependent data. The paper outlines our estimation techniques, discusses convergence rates, establishes a lower bound on the minimax risk, and demonstrates the asymptotic normality of the estimator for $f_X$ under smooth error distributions. Through examples and simulations, we showcase the efficacy of our estimator. The paper concludes by providing proofs for the presented theoretical results.v

翻译：本文聚焦于在乘法测量误差模型$Y_t = X_t U_t$中估计强混合平稳过程$X_t$的不变密度函数$f_X$，其中$U_t$也是强混合平稳过程。我们提出了一种处理非独立数据的新方法，该方法在现实场景中具有典型性。例如，从不同群体收集的数据可能在群体内部存在相互依赖，类似于$m$相依平稳过程生成的数据——该过程属于平稳过程的一个子类。本研究将模型$Y_t = X_t U_t$的适用性扩展至处理复杂依赖数据的多个科学领域。本文阐述了估计技术，讨论了收敛速率，建立了极小化风险的下界，并证明了在光滑误差分布下$f_X$估计量的渐近正态性。通过实例与仿真，我们展示了估计量的有效性。文末给出了相关理论结果的证明。