Limits of Some Weighted Cesaro Averages

Crismale, Vitonofrio; Fidaleo, Francesco; Lu, Yun Gang

doi:10.1007/s00025-016-0622-z

Cited by 2 publications

(1 citation statement)

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“…x n where x n ∈ R for n ∈ N (here and throughout this paper, N denotes the natural numbers excluding 0). These are known as Cesàro limits (see [1] for example) or sometimes as Cesàro means or Cesàro averages (as in [2]), and arise naturally in multiple mathematical fields, including statistics, probability, functional and general analysis, and in the study of stochastic processes, particularly ergodic processes. They are important in many applications (see list of references in [1]).…”

Section: Introductionmentioning

confidence: 99%

Binary sequences with a Cesàro limit

Keith¹,

Markowsky²

2021

Preprint

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The Cesàro limit -the asymptotic average of a sequence of real numbers -is an operator of fundamental importance in probability, statistics and mathematical analysis. To better understand sequences with Cesàro limits, this paper considers the space F comprised of all binary sequences with a Cesàro limit, and the associated functional ν : F → [0, 1] mapping each such sequence to its Cesàro limit. The basic properties of F and ν are enumerated, and chains (totally ordered sets) in F on which ν is countably additive are studied in detail. The main result of the paper concerns a structural property of the pair (F , ν), specifically that F can be factored (in a certain sense) to produce a monotone class on which ν is countably additive. In the process, a slight generalisation and clarification of the monotone class theorem for Boolean algebras is proved.

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Section: Introductionmentioning

confidence: 99%