We prove that certain asymptotic moments exist for some random distance expanding dynamical systems and Markov chains in random dynamical environment, and compute them in terms of the derivatives at 0 of an appropriate pressure function. It will follow that these moments satisfy the relations that the asymptotic moments $${\gamma }_k=\lim _{n\rightarrow \infty }n^{-[\frac{k}{2}]}{\mathbb {E}}(\sum _{i=1}^n X_i)^k$$ γ k = lim n → ∞ n - [ k 2 ] E ( ∑ i = 1 n X i ) k of sums of independent and identically distributed centered random variables satisfy. Under certain mixing conditions we will also estimate the convergence rate towards these limits. The arguments in the proof of these results yield that the partial sums generated by the random Ruelle–Perron–Frobenius triplets and all of their parametric derivatives (considered as functions on the base) corresponding to appropriate random transfer or Markov operators satisfy several probabilistic limit theorems such as the central limit theorem. We will also obtain certain (Edgeworth) asymptotic expansions related to the central limit theorem for such processes. Our proofs rely on a (parametric) random complex Ruelle-Perron-Frobenius theorem, which replaces some of the spectral techniques used in literature in order to obtain limit theorems for deterministic dynamical systems and Markov chains.

中文翻译：

---

关于随机动力环境中某些过程的渐近矩和埃奇沃斯展开

我们证明了在随机动力环境中某些随机距离扩展动力系统和马尔可夫链存在某些渐近矩，并根据适当压力函​​数在 0 处的导数来计算它们。可以得出，这些矩满足渐近矩 $${\gamma }_k=\lim _{n\rightarrow \infty }n^{-[\frac{k}{2}]}{\mathbb { E}}(\sum _{i=1}^n X_i)^k$$ γ k = lim n → ∞ n - [ k 2 ] E ( ∑ i = 1 n X i ) k 个独立相同的和分布式中心随机变量满足。在某些混合条件下，我们还将估计朝向这些限制的收敛速度。证明这些结果的论据产生由随机 Ruelle-Perron-Frobenius 三元组及其所有参数导数（被视为基函数）对应于适当的随机转移或马尔可夫算子生成的部分和满足几个概率极限定理比如中心极限定理。我们还将获得与此类过程的中心极限定理相关的某些（埃奇沃斯）渐近展开式。我们的证明依赖于（参数）随机复数 Ruelle-Perron-Frobenius 定理，该定理取代了文献中使用的一些频谱技术，以获得确定性动力系统和马尔可夫链的极限定理。我们还将获得与此类过程的中心极限定理相关的某些（埃奇沃斯）渐近展开式。我们的证明依赖于（参数）随机复数 Ruelle-Perron-Frobenius 定理，该定理取代了文献中使用的一些谱技术，以获得确定性动力系统和马尔可夫链的极限定理。我们还将获得与此类过程的中心极限定理相关的某些（埃奇沃斯）渐近展开式。我们的证明依赖于（参数）随机复数 Ruelle-Perron-Frobenius 定理，该定理取代了文献中使用的一些频谱技术，以获得确定性动力系统和马尔可夫链的极限定理。