Stochastic Approximation (SA) was introduced in the early 1950's and has been an active area of research for several decades. While the initial focus was on statistical questions, it was seen to have applications to signal processing, convex optimisation. %Over the last decade, there has been a revival of interest in SA as In later years SA has found application in Reinforced Learning (RL) and led to revival of interest. While bulk of the literature is on SA for the case when the observations are from a finite dimensional Euclidian space, there has been interest in extending the same to infinite dimension. Extension to Hilbert spaces is relatively easier to do, but this is not so when we come to a Banach space - since in the case of a Banach space, even {\em law of large numbers} is not true in general. We consider some cases where approximation works in a Banach space. Our framework includes case when the Banach space $\Bb$ is $\Cb([0,1],\R^d)$, as well as $\L^1([0,1],\R^d)$, the two cases which do not even have the Radon-Nikodym property.

翻译：随机逼近（Stochastic Approximation, SA）于20世纪50年代初提出，数十年来一直是活跃的研究领域。最初的研究重点集中于统计问题，随后发现其在信号处理、凸优化等领域具有应用价值。近年来，SA在强化学习（Reinforced Learning, RL）中得到应用，并重新激发了研究兴趣。尽管现有文献主要关注观测数据来自有限维欧几里得空间的情形，但将其推广至无限维空间的研究也逐渐兴起。推广至希尔伯特空间相对容易实现，但涉及巴拿赫空间时则面临挑战——因为在巴拿赫空间中，即便是一般的强大数律也难以成立。本文探讨了在巴拿赫空间中能够实现逼近的若干情形，其研究框架涵盖巴拿赫空间$\Bb$为$\Cb([0,1],\R^d)$和$\L^1([0,1],\R^d)$两种情况，这两种空间甚至不具备拉东-尼科迪姆性质。