On Exponential Convergence of Random Variables

TL;DR

This paper investigates the relationships between exponential convergence of expectations, trajectories, and hitting times in nonnegative random variables, with applications in optimization, control, and estimation.

Contribution

It provides a general theoretical framework linking different types of exponential convergence and demonstrates its relevance in various applied fields.

Findings

Established conditions for exponential convergence of expectations and trajectories.

Linked exponential convergence with the growth rate of expected hitting times.

Applied theoretical results to optimization, stochastic control, and estimation.

Abstract

Given the discrete-time sequence of nonnegative random variables, general dependencies between the exponential convergence of the expectations, exponential convergence of the trajectories and the logarithmic growth of the corresponding expected hitting times are analysed. The applications are presented: the general results are applied to the areas of optimization, stochastic control and estimation.