Let $(\mathcal{U},d)$ be a complete separable metric space and ${\left(F_n\right)}_{n\ge 0}$ a sequence of random functions from $\mathcal{U}$ to $\mathcal{U}$. Motivated by studying the stability property for Markovian dynamic models, in this paper, we assume that the random function ${\left(F_n\right)}_{n\ge 0}$ is driven by a Markov chain $\mathbf{X}=\{X_n,n\ge 0\}$. Under some regularity conditions on the driving Markov chain and the mean contraction assumption, we show that the forward iterations $M_n^u=F_n\circ \cdots \circ F_1\left(u\right)$, $n\ge 0$, converge weakly to a unique stationary distribution $\Pi$ for each $u\in \mathcal{U}$, where $\circ$ denotes composition of two maps. The associated backward iterations ${\tilde{M}}_n^u=F_1\circ \cdots \circ F_n\left(u\right)$ are almost surely convergent to a random variable ${\tilde{M}}_{\infty }$ which does not depend on $u$ and has distribution $\Pi$. Moreover, under suitable moment conditions, we provide estimates and rate of convergence for $d({\tilde{M}}_{\infty },{\tilde{M}}_n^u)$ and $d(M_n^u,M_n^v)$, $u,v\in \mathcal{U}$. The results are applied to the examples that have been discussed in the literature, including random coefficient autoregression models and recurrent neural network.

让 $（\mathcal{ü}，d）$ 是一个完全可分离的度量空间， ${（F_{ñ}）}_{ñ\ge 0}$ 来自的随机函数序列 $\mathcal{ü}$ 至 $\mathcal{ü}$。通过研究马尔可夫动力学模型的稳定性，本文假设随机函数${（F_{ñ}）}_{ñ\ge 0}$ 由马尔可夫链驱动 $\mathbf{X}=\{X_{ñ}，ñ\ge 0\}$。在驱动马尔可夫链的某些规律性条件下和平均收缩假设下，我们证明了正向迭代${\mathrm{中号}}_{ñ}^{ü}=F_{ñ}\circ \cdots \circ F_{\mathrm{1个}}（ü）$， $ñ\ge 0$，微弱地收敛到唯一的平稳分布 $\Pi$ 对于每个 $ü\in \mathcal{ü}$， 在哪里 $\circ$表示两个地图的组成。相关的向后迭代${\widetilde{\mathrm{中号}}}_{ñ}^{ü}=F_{\mathrm{1个}}\circ \cdots \circ F_{ñ}（ü）$ 几乎可以肯定地收敛于随机变量 ${\widetilde{\mathrm{中号}}}_{\infty }$ 不依赖于 $ü$ 并有分布 $\Pi$。此外，在合适的时刻条件下，我们提供了以下估计和收敛速度：$d（{\widetilde{\mathrm{中号}}}_{\infty }，{\widetilde{\mathrm{中号}}}_{ñ}^{ü}）$ 和 $d（{\mathrm{中号}}_{ñ}^{ü}，{\mathrm{中号}}_{ñ}^v）$， $ü，v\in \mathcal{ü}$。将结果应用于文献中已讨论的示例，包括随机系数自回归模型和递归神经网络。