Multivariable generalizations of bivariate means via invariance

2024/02/06 by Paweł Pasteczka, Pasteczka, Paweł
Mathematics · #Advanced Statistical Methods and Models #Dynamical Systems (math.DS) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2402.04121

openalex publication_date 2024/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a given p-variable mean M \colon I^p → I (I is a subinterval of ℝ), following (Horwitz, 2002) and (Lawson and Lim, 2008), we can define (under certain assumption) its (p+1)-variable β-invariant extension as the unique solution K \colon I^p+1 → I of the functional equation K(M(x₂,…,x_p+1)amp;,M(x₁,x₃,…,x_p+1),…,M(x₁,…,x_p))
amp;=K(x₁,…,x_p+1), for all x₁,…,x_p+1 ∈ I in the family of means. Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions.

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