Observable operator models (OOMs) offer a powerful framework for modelling stochastic processes, surpassing the traditional hidden Markov models (HMMs) in generality and efficiency. However, using OOMs to model infinite-dimensional processes poses significant theoretical challenges. This article explores a rigorous approach to developing an approximation theory for OOMs of infinite-dimensional processes. Building upon foundational work outlined in an unpublished tutorial [Jae98], an inner product structure on the space of future distributions is rigorously established and the continuity of observable operators with respect to the associated 2-norm is proven. The original theorem proven in this thesis describes a fundamental obstacle in making an infinite-dimensional space of future distributions into a Hilbert space. The presented findings lay the groundwork for future research in approximating observable operators of infinite-dimensional processes, while a remedy to the encountered obstacle is suggested.

翻译：可观测算子模型（OOMs）为随机过程建模提供了强大框架，在泛化性和效率上超越了传统隐马尔可夫模型（HMMs）。然而，使用OOMs对无限维过程建模存在显著的理论挑战。本文探索了发展无限维过程OOMs逼近理论的严谨方法。基于未发表教程[Jae98]中的基础性工作，本文严格建立了未来分布空间上的内积结构，并证明了可观测算子关于关联的2-范数的连续性。本文提出的原始定理揭示了将无限维未来分布空间构建为希尔伯特空间的基本障碍。所述发现为未来研究无限维过程可观测算子逼近奠定了基础，同时针对所遇障碍提出了解决方案。