Exponential inequalities for Mann's iterative scheme with functional random errors

TL;DR

This paper analyzes the convergence behavior of Mann's iterative scheme with functional random errors, providing exponential inequalities, convergence rates, and confidence sets for fixed point approximation.

Contribution

It introduces exponential inequalities and confidence sets for Mann's iteration under functional random errors, advancing understanding of its convergence properties.

Findings

Almost complete convergence to the fixed point is established.

Explicit rate of convergence is derived.

A confidence set for the fixed point is constructed.

Abstract

In this work, we deal with an iteration method for approximating a fixed point of a contraction mapping using the Mann's algorithm under functional random errors. We first show its almost complete convergence to the fixed point by mean of an exponential inequality and then we specify the induced rate of convergence. We finally build a confidence set for the fixed point.