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 ${\{M^{d}:d \in { Dyad} \left([0,1]\right) \}}$ parameterized by ${{Dyad} \left([0,1]\right) }$, 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 ${\{M^{d}(x, y):d \in {Dyad} \left([0, 1]\right) \}}$ 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.