This work advances the theoretical foundations of reservoir computing (RC) by providing a unified treatment of fading memory and the echo state property (ESP) in both deterministic and stochastic settings. We investigate state-space systems, a central model class in time series learning, and establish that fading memory and solution stability hold generically -- even in the absence of the ESP -- offering a robust explanation for the empirical success of RC models without strict contractivity conditions. In the stochastic case, we critically assess stochastic echo states, proposing a novel distributional perspective rooted in attractor dynamics on the space of probability distributions, which leads to a rich and coherent theory. Our results extend and generalize previous work on non-autonomous dynamical systems, offering new insights into causality, stability, and memory in RC models. This lays the groundwork for reliable generative modeling of temporal data in both deterministic and stochastic regimes.

翻译：本研究通过统一处理确定性与随机性背景下的衰减记忆和回声状态特性，深化了储层计算的理论基础。我们研究了时间序列学习的核心模型类别——状态空间系统，并证明衰减记忆与解稳定性在一般情况下均成立——即使在没有回声状态特性的条件下——这为储层计算模型在缺乏严格压缩性条件下的实证成功提供了稳健的解释。在随机性情形中，我们批判性地评估了随机回声状态，提出了一种植根于概率分布空间上吸引子动力学的新分布视角，从而发展出一套丰富而自洽的理论。我们的结果扩展并推广了先前关于非自治动力系统的工作，为储层计算模型中的因果性、稳定性与记忆机制提供了新的见解。这为在确定性与随机性机制下实现时间数据的可靠生成建模奠定了理论基础。