Convergence of sequences: a survey

TL;DR

This survey reviews the literature on convergence theorems for sequences of real numbers, emphasizing their importance in system theory, optimization, and game theory, and explores their connection with Fejer monotonicity in various settings.

Contribution

It provides a comprehensive overview of convergence theorems and their relation to Fejer monotonicity in both deterministic and stochastic contexts.

Findings

Summarizes key convergence theorems in the literature.

Explores the role of Fejer monotonicity in convergence analysis.

Connects theoretical results with practical applications in system and optimization theories.

Abstract

Convergent sequences of real numbers play a fundamental role in many different problems in system theory, e.g., in Lyapunov stability analysis, as well as in optimization theory and computational game theory. In this survey, we provide an overview of the literature on convergence theorems and their connection with Fejer monotonicity in the deterministic and stochastic settings, and we show how to exploit these results.