On the rate of convergence of Krasnoselski-Mann iterations and their   connection with sums of Bernoullis
  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. 
Roberto Cominetti, Jos\'e A. Soto, Jos\'e Vaisman
math.OC math.FA math.NA math.PR