On the new smoothness class of means and its impact to mean-type mappings

TL;DR

This paper introduces residual means with specific Taylor expansion properties, classifies symmetric means as residual, and analyzes the asymptotic behavior of variance sequences under mean-type mappings.

Contribution

It defines residual means with a new smoothness class, proves all thrice differentiable symmetric means are residual, and applies this to study variance limits in mean iterations.

Findings

All symmetric three-times differentiable means are residual.

Calculated residuum for quasideviation means.

Established limit behavior of variance sequences under residual mean mappings.

Abstract

We define so-called residual means, which have a Taylor expansion of the form $M(x)=\bar x +\tfrac12 \xi_M(\bar x) \text{Var}(x)+o(\|x-\bar x\|^\alpha)$ for some $\alpha>2$ and a single-variable function $\xi_M$ ($\bar x$ stands for the arithmetic mean of the vector $x$), and show that all symmetric means which are three times continuously differentiable are residual. We also calculate the value of residuum for quasideviation means and a few subclasses of this family. Later, we apply it to establish the limit of the sequence $\big(\frac{\text{Var}\ {\bf M}^{n+1}(x)}{(\text{Var}\ {\bf M}^n(x))^2}\big)_{n=1}^\infty$, where ${\bf M} \colon I^p\to I^p$ is a mean-type mapping consisting of $p$-variable residual means on an interval $I$, and $x \in I^p$ is a nonconstant vector.