Weighting by Iteration: The Case of Ryll-Nardzewski’s Iterations

Abstract

Aczél’s and Ryll-Nardzewski’s dyadic iterations are iterative procedures which associate to a given mean M a family of means { Md: d∈ Dyad ([ 0 , 1 ]) } parameterized by Dyad ([ 0 , 1 ]) , the dyadic fractions of the interval [0, 1]. Aczél’s iterations exhibit a nice characteristic: when M is a strict continuous mean and x < y, the set { Md(x, y) : d∈ Dyad ([ 0 , 1 ]) } is dense in [x, y]. This fact is in the basis of the construction of an algorithm of weighting for an ample class of means. In pursuit of a similar algorithm using Ryll-Nardzewski’s instead of Aczél’s iterations, a series of obstacles is found, which motivates the detailed study of these last conducted along this paper. Among other result of interest, several conditions on the mean M are identified which make viable a weighting algorithm based on these iterations.