A probabilistic framework to study the dependence structure induced by deterministic discrete-time state-space systems between input and output processes is introduced. General sufficient conditions are formulated under which output processes exist and are unique once an input process has been fixed, a property that in the deterministic state-space literature is known as the echo state property. When those conditions are satisfied, the given state-space system becomes a generative model for probabilistic dependences between two sequence spaces. Moreover, those conditions guarantee that the output depends continuously on the input when using the Wasserstein metric. The output processes whose existence is proved are shown to be causal in a specific sense and to generalize those studied in purely deterministic situations. The results in this paper constitute a significant stochastic generalization of sufficient conditions for the deterministic echo state property to hold, in the sense that the stochastic echo state property can be satisfied under contractivity conditions that are strictly weaker than those in deterministic situations. This means that state-space systems can induce a purely probabilistic dependence structure between input and output sequence spaces even when there is no functional relation between those two spaces.

翻译：本文引入了一个概率框架，用于研究确定性离散时间状态空间系统在输入与输出过程之间所诱导的依赖结构。我们构建了充分的一般性条件，在这些条件下，一旦输入过程确定，输出过程便存在且唯一——这一性质在确定性状态空间文献中被称为回声状态特性。当这些条件满足时，给定的状态空间系统便成为两个序列空间之间概率依赖关系的生成模型。此外，这些条件保证了在使用Wasserstein度量时，输出对输入的连续依赖性。所证明存在的输出过程在特定意义上是因果性的，并推广了纯确定性情形下所研究的过程。本文的结果构成了对确定性回声状态特性充分条件的显著随机推广，其意义在于：随机回声状态特性可以在比确定性情形更弱的压缩性条件下得到满足。这意味着，即使输入与输出序列空间之间不存在函数关系，状态空间系统仍能在它们之间诱导出纯粹的概率依赖结构。