We prove central limit theorems, Berry-Esseen type theorems, almost sure invariance principles, large deviations and Livsic type regularity for partial sums of the form $S_n=\sum_{j=0}^{n-1}f_j(...,X_{j-1},X_j,X_{j+1},...)$, where $(X_j)$ is an inhomogeneous Markov chain satisfying some mixing assumptions and $f_j$ is a sequence of sufficiently regular functions. Even though the case of non-stationary chains and time dependent functions $f_j$ is more challenging, our results seem to be new already for stationary Markov chains. They also seem to be new for non-stationary Bernoulli shifts (that is when $(X_j)$ are independent but not identically distributed). This paper is the first one in a series of two papers. In \cite{Work} we will prove local limit theorems including developing the related reduction theory in the sense of \cite{DolgHaf LLT, DS}.

翻译：我们证明了形如 $S_n=\sum_{j=0}^{n-1}f_j(...,X_{j-1},X_j,X_{j+1},...)$ 的部分和满足中心极限定理、Berry-Esseen型定理、几乎必然不变原理、大偏差及Livsic型正则性，其中 $(X_j)$ 是满足特定混合条件的非齐次马尔可夫链，$f_j$ 是一组充分正则的函数序列。尽管非平稳链与时间依赖函数 $f_j$ 的情形更具挑战性，但我们的结果即使对于平稳马尔可夫链而言似乎也是全新的。这些结果对于非平稳伯努利推移（即 $(X_j)$ 独立但非同分布的情形）同样具有新颖性。本文是由两篇论文组成的系列研究中的首篇。在文献 \cite{Work} 中，我们将证明包括按 \cite{DolgHaf LLT, DS} 意义发展的相关约化理论在内的局部极限定理。