In the context of stochastic approximation algorithms with more general update rules, such as those involving Martingale difference sequences or other additive noise terms, the stability and convergence results can be extended by considering the impact of these additional noise terms on the iterative process.
For stochastic approximation algorithms with Martingale difference sequences, the key lies in analyzing the behavior of the iterates in the presence of this specific type of noise. By incorporating the properties of Martingale sequences into the stability analysis, one can establish conditions under which the iterates remain bounded almost surely, leading to convergence results similar to those obtained in the original work.
When dealing with other additive noise terms, the extension of stability and convergence results involves understanding how these additional noise components affect the overall behavior of the algorithm. By incorporating the properties and characteristics of the specific noise terms into the analysis, one can derive conditions for stability and convergence that account for the presence of such noise in the update rules.
Overall, the extension to stochastic approximation algorithms with more general update rules requires a thorough analysis of how different types of noise impact the iterative process and the overall behavior of the algorithm. By adapting the stability and convergence analysis to accommodate these variations, one can provide a comprehensive framework for analyzing the performance of algorithms in the presence of diverse noise sources.