Strongly mixed random errors in Mann's iteration algorithm for a contractive real function

TL;DR

This paper investigates Mann's stochastic iteration algorithm with strong mixing errors, establishing convergence rates and confidence intervals, supported by numerical examples including an astronomy case.

Contribution

It introduces Fuk-Nagaev inequalities for strong mixing errors in Mann's iteration, providing convergence rates and confidence interval construction.

Findings

Proved almost complete convergence with explicit rates.

Derived confidence intervals for fixed points.

Validated results with numerical astronomy example.

Abstract

This work deals with the Mann's stochastic iteration algorithm under strong mixing random errors. We establish the Fuk-Nagaev's inequalities that enable us to prove the almost complete convergence with its corresponding rate of convergence. Moreover, these inequalities give us the possibility of constructing a confidence interval for the unique fixed point. Finally, to check the feasibility and validity of our theoretical results, we consider some numerical examples, namely a classical example from astronomy.