We consider time-dependent dynamical systems arising as sequential compositions of self-maps of a probability space. We establish conditions under which the Birkhoff sums for multivariate observations, given a centering and a general normalizing sequence $b(N)$ of invertible square matrices, are approximated by a normal distribution with respect to a metric of regular test functions. Depending on the metric and the normalizing sequence $b(N)$, the conditions imply that the error in the approximation decays either at the rate $O(N^{-1/2})$ or the rate $O(N^{-1/2} \log N)$, under the additional assumption that $\Vert b(N)^{-1} \Vert \lesssim N^{-1/2}$. The error comes with a multiplicative constant whose exact value can be computed directly from the conditions. The proof is based on an observation due to Sunklodas regarding Stein's method of normal approximation. We give applications to one-dimensional random piecewise expanding maps and to sequential, random, and quasistatic intermittent systems.

中文翻译：

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Sunklodas 的瞬态动力系统正态逼近方法

我们认为依赖于时间的动态系统是概率空间自映射的顺序组合。我们建立了条件，在该条件下，给定可逆方阵的居中和一般归一化序列 $b(N)$ 的多元观测值的 Birkhoff 总和通过关于常规测试函数的度量的正态分布近似。根据度量和归一化序列 $b(N)$，条件意味着近似误差以 $O(N^{-1/2})$ 或 $O(N^ {-1/2} \log N)$，在额外假设 $\Vert b(N)^{-1} \Vert \lesssim N^{-1/2}$ 下。该错误带有一个乘法常数，其确切值可以直接从条件中计算出来。该证明基于 Sunklodas 对 Stein' 的观察 s 正态逼近方法。我们将应用应用于一维随机分段扩展映射以及顺序、随机和准静态间歇系统。