In two-time-scale stochastic approximation (SA), two iterates are updated at different rates, governed by distinct step sizes, with each update influencing the other. Previous studies have demonstrated that the convergence rates of the error terms for these updates depend solely on their respective step sizes, a property known as decoupled convergence. However, a functional version of this decoupled convergence has not been explored. Our work fills this gap by establishing decoupled functional central limit theorems for two-time-scale SA, offering a more precise characterization of its asymptotic behavior. Our results show that, on each time scale, the limiting dynamics has the same form as in standard SA, and the coupling between the two iterates enters the limit only through the associated coefficients. To achieve these results, we leverage the martingale problem approach and establish tightness as a crucial intermediate step. Furthermore, to address the interdependence between different time scales, we introduce an innovative auxiliary sequence to eliminate the primary influence of the fast-time-scale update on the slow-time-scale update.

翻译：在双时间尺度随机逼近（SA）中，两个迭代变量以不同的速率更新，由不同的步长控制，且每次更新相互影响。先前的研究已证明，这些更新误差项的收敛速率仅取决于其各自的步长，这一性质被称为解耦收敛。然而，这种解耦收敛的泛函版本尚未被探索。我们的工作通过建立双时间尺度SA的解耦泛函中心极限定理填补了这一空白，为其渐近行为提供了更精确的描述。我们的结果表明，在每个时间尺度上，极限动力学具有与标准SA相同的形式，且两个迭代变量之间的耦合仅通过相关系数进入极限。为了获得这些结果，我们利用了鞅问题方法，并将紧致性确立为关键中间步骤。此外，为了处理不同时间尺度间的相互依赖关系，我们引入了一个创新的辅助序列，以消除快时间尺度更新对慢时间尺度更新的主要影响。