A finitary Kronecker's lemma and large deviations in the strong law of large numbers on Banach spaces

IF 0.6 2区 数学 Q2 LOGIC
Morenikeji Neri
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引用次数: 0

Abstract

We explore the computational content of Kronecker's lemma via the proof-theoretic perspective of proof mining and utilise the resulting finitary variant of this fundamental result to provide new rates for the Strong Law of Large Numbers for random variables taking values in type p Banach spaces, which in particular are very uniform in the sense that they do not depend on the distribution of the random variables. Furthermore, we provide computability-theoretic arguments to demonstrate the ineffectiveness of Kronecker's lemma and investigate the result from the perspective of Reverse Mathematics. In addition, we demonstrate how this ineffectiveness from Kronecker's lemma trickles down to the Strong Law of Large Numbers by providing a construction that shows that computable rates of convergence are not always possible. Lastly, we demonstrate how Kronecker's lemma falls under a class of deterministic formulas whose solution to their Dialectica interpretation satisfies a continuity property and how, for such formulas, one obtains an upgrade principle that allows one to lift computational interpretations of deterministic results to quantitative results for their probabilistic analogue. This result generalises the previous work of the author and Pischke.
我们通过证明挖掘的证明论视角探索克罗内克两难的计算内容,并利用这一基本结果的有限变体,为在 p 型巴拿赫空间取值的随机变量的强大数定律提供新的速率,特别是在不依赖于随机变量分布的意义上,这种速率是非常均匀的。此外,我们还提供了可计算性理论论据来证明克罗内克∞的无效性,并从逆数学的角度研究了这一结果。此外,我们还提供了一个构造,说明可计算的收敛率并不总是可能的,以此证明克罗内克两难的无效性是如何向下渗透到大数强律的。最后,我们证明了克罗内克两难如何属于一类确定性公式,其辩证解释的解满足连续性属性,以及对于这类公式,我们如何获得一个升级原理,允许我们将确定性结果的计算解释提升为其概率类似的定量结果。这一结果概括了作者和皮施克之前的工作。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.40
自引率
12.50%
发文量
78
审稿时长
200 days
期刊介绍: The journal Annals of Pure and Applied Logic publishes high quality papers in all areas of mathematical logic as well as applications of logic in mathematics, in theoretical computer science and in other related disciplines. All submissions to the journal should be mathematically correct, well written (preferably in English)and contain relevant new results that are of significant interest to a substantial number of logicians. The journal also considers submissions that are somewhat too long to be published by other journals while being too short to form a separate memoir provided that they are of particular outstanding quality and broad interest. In addition, Annals of Pure and Applied Logic occasionally publishes special issues of selected papers from well-chosen conferences in pure and applied logic.
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