A stochastic process that arises by composing a function with a Markov process is called an aggregated Markov process (AMP). The purpose of composing a Markov process with a function can be a reduction of dimensions, e.g., a projection onto certain coordinates. The theory around AMP has been extensively studied e.g. by Dynkin, Cameron, Rogers and Pitman, and Kelly, all of whom provided sufficient conditions for an AMP to remain Markov. In another direction, Larget provided a canonical representation for AMP, which can be used to verify the equivalence of two AMPs. The purpose of this paper is to describe how the theory of AMP can be applied to stochastic learning theory as they learn a particular task.

翻译：通过将函数与马尔可夫过程复合而得到的随机过程称为聚合马尔可夫过程（AMP）。对马尔可夫过程进行函数复合的目的可以是降维（例如投影到特定坐标）。围绕AMP的理论已被Dynkin、Cameron、Rogers和Pitman以及Kelly等人广泛研究，他们均给出了AMP保持马尔可夫性的充分条件。在另一个方向上，Larget提出了AMP的规范表示方法，可用于验证两个AMP的等价性。本文旨在描述如何将AMP理论应用于随机学习理论中，以解决特定任务的学习过程。