对Choquet积分的Chebyshev不等式的重新考察

IF 2.7 1区数学 Q2 COMPUTER SCIENCE, THEORY & METHODS

Fuzzy Sets and Systems Pub Date : 2025-10-02 DOI:10.1016/j.fss.2025.109608

Hamzeh Agahi

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引用次数: 0

摘要

最近，Wang [43, Fuzzy Sets and Systems 457,(2023) 105-118]提出Choquet积分的Chebyshev不等式在共单调或超模性下成立。我们的反例表明，对于无共单调性的Choquet积分，单靠超模性不能证明Chebyshev不等式。我们还为Choquet积分建立了一个改进的Chebyshev不等式，它提供了更严格的界。作为概率论中的一个应用，我们导出了一个增强的方差不等式，使伯努利分布达到相等。最后，我们提出了一个有待进一步研究的开放性问题。

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Chebyshev's inequality for Choquet integral revisited

Recently, Wang [43, Fuzzy Sets and Systems 457, (2023) 105–118] claimed that Chebyshev's inequality for Choquet integral holds under comonotonicity or supermodularity. Our counter-example demonstrates that supermodularity alone cannot validate Chebyshev's inequality for Choquet integral without comonotonicity. We also establish a refined Chebyshev inequality for Choquet integral that provides tighter bounds. As an application in probability theory, we derive a strengthened variance inequality that achieves equality for Bernoulli distributions. Finally, we present an open problem for future research.

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来源期刊

Fuzzy Sets and Systems 数学-计算机：理论方法

CiteScore

6.50

自引率

17.90%

发文量

321

审稿时长

6.1 months

期刊介绍： Since its launching in 1978, the journal Fuzzy Sets and Systems has been devoted to the international advancement of the theory and application of fuzzy sets and systems. The theory of fuzzy sets now encompasses a well organized corpus of basic notions including (and not restricted to) aggregation operations, a generalized theory of relations, specific measures of information content, a calculus of fuzzy numbers. Fuzzy sets are also the cornerstone of a non-additive uncertainty theory, namely possibility theory, and of a versatile tool for both linguistic and numerical modeling: fuzzy rule-based systems. Numerous works now combine fuzzy concepts with other scientific disciplines as well as modern technologies. In mathematics fuzzy sets have triggered new research topics in connection with category theory, topology, algebra, analysis. Fuzzy sets are also part of a recent trend in the study of generalized measures and integrals, and are combined with statistical methods. Furthermore, fuzzy sets have strong logical underpinnings in the tradition of many-valued logics.