Quasi-arithmetic-type invariant means on probability space

Derȩgowska, Beata; Pasteczka, Paweł

doi:10.1007/s00010-020-00765-8

Cited by 1 publication

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“…This result has several generalisations (see, for example, Matkowski [33], and Matkowski-Pasteczka [35]). For details we refer the reader to the reach literature on the subject, the classical one Lagrange [25], Gauss [16], Foster-Philips [15], Lehmer [27], Schoenberg [44] as well as more recent Baják-Páles [3][4][5][6], Daróczy-Páles [9,11,12], Deręgowska-Pasteczka [14], Głazowska [18,19], Jarczyk-Jarczyk [22], Matkowski [30][31][32][33], Matkowski-Páles [36], Matkowski-Pasteczka [34,35] and Pasteczka [38,39,41,42].…”

Section: Introductionmentioning

confidence: 99%

Invariance Property for Extended Means

Pasteczka

2023

Results Math

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We study the properties of the mean-type mappings $$\textbf{M}:I^p \rightarrow I^p$$ M : I p → I p of the form $$\begin{aligned} \textbf{M}(x_1,\dots ,x_p):=\big (M_1(x_{\alpha _{1,1}},\dots ,x_{\alpha _{1,d_1}}), \dots ,M_p(x_{\alpha _{p,1}},\dots ,x_{\alpha _{p,d_p}})\big ), \end{aligned}$$ M ( x 1 , ⋯ , x p ) : = ( M 1 ( x α 1 , 1 , ⋯ , x α 1 , d 1 ) , ⋯ , M p ( x α p , 1 , ⋯ , x α p , d p ) ) , where p and $$d_i$$ d i -s are positive integers, each $$M_i$$ M i is a $$d_i$$ d i -variable mean on an interval $$I \subset {\mathbb {R}}$$ I ⊂ R , and $$\alpha _{i,j}$$ α i , j -s are elements from $$\{1,\dots ,p\}$$ { 1 , ⋯ , p } . We show that, under some natural assumption on $$M_i$$ M i -s, the problem of existing the unique $$\textbf{M}$$ M -invariant mean can be reduced to the ergodicity of the directed graph with vertexes $$\{1,\dots ,p\}$$ { 1 , ⋯ , p } and edges $$\{(\alpha _{i,j},i) :i,j \text { admissible}\}$$ { ( α i , j , i ) : i , j admissible } .

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