Binary sequences with a Ces\`aro limit
| Autor: | Keith, Jonathan M., Markowsky, Greg |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | The Ces\`aro 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 Ces\`aro limits, this paper considers the space $\mathcal{F}$ comprised of all binary sequences with a Ces\`aro limit, and the associated functional $\nu: \mathcal{F} \rightarrow [0,1]$ mapping each such sequence to its Ces\`aro limit. The basic properties of $\mathcal{F}$ and $\nu$ are enumerated, and chains (totally ordered sets) in $\mathcal{F}$ on which $\nu$ is countably additive are studied in detail. The main result of the paper concerns a structural property of the pair $(\mathcal{F},\nu)$, specifically that $\mathcal{F}$ can be factored (in a certain sense) to produce a monotone class on which $\nu$ is countably additive. In the process, a slight generalisation and clarification of the monotone class theorem for Boolean algebras is proved. Comment: arXiv admin note: substantial text overlap with arXiv:2104.08705 |
| Databáze: | arXiv |
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