On the rate of convergence of Krasnoselski-Mann iterations and their connection with sums of Bernoullis

TL;DR

This paper provides a new estimate for the convergence rate of Krasnoselski-Mann iterations, confirming a conjecture and linking the process to Bernoulli sums and Poisson bounds.

Contribution

It establishes the asymptotic regularity of Krasnoselski-Mann iterations and introduces a novel approach connecting these iterations with stochastic Bernoulli processes.

Findings

Confirmed the Baillon-Bruck conjecture on asymptotic regularity.

Derived a new Hoeffding-type inequality for Bernoulli sums.

Connected fixed point iteration convergence with stochastic process analysis.

Abstract

In this paper we establish an estimate for the rate of convergence of the Krasnosel'ski\v{\i}-Mann iteration for computing fixed points of non-expansive maps. Our main result settles the Baillon-Bruck conjecture [3] on the asymptotic regularity of this iteration. The proof proceeds by establishing a connection between these iterates and a stochastic process involving sums of non-homogeneous Bernoulli trials. We also exploit a new Hoeffding-type inequality to majorize the expected value of a convex function of these sums using Poisson distributions.