Invariance principles for the law of the iterated logarithm under G-framework

Wu, Panyu; Chen, ZengJing

doi:10.1007/s11425-015-5002-8

Cited by 13 publications

(6 citation statements)

References 17 publications

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“…He also obtained the Hartman-Wintner’s law of iterated logarithm and Kolmogorov’s strong law of large numbers for identically distributed and extended negatively dependent random variables. Wu and Chen [ 10 ] also researched the law of the iterated logarithm, and Cheng [ 11 ] studied the strong law of larger number with a general moment condition , and so on. Many powerful inequations and conventional methods for linear expectation and probabilities are no longer valid, the study of limit theorems under sub-linear expectation becomes much more challenging.…”

Section: Introductionmentioning

confidence: 99%

Complete convergence and complete moment convergence for weighted sums of extended negatively dependent random variables under sub-linear expectation

Zhong

2017

J Inequal Appl

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In this paper, we study the complete convergence and complete moment convergence for weighted sums of extended negatively dependent (END) random variables under sub-linear expectations space with the condition of , further , ( is a slow varying and monotone nondecreasing function). As an application, the Baum-Katz type result for weighted sums of extended negatively dependent random variables is established under sub-linear expectations space. The results obtained in the article are the extensions of the complete convergence and complete moment convergence under classical linear expectation space.

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Section: Introductionmentioning

confidence: 99%

Complete convergence and complete moment convergence for weighted sums of extended negatively dependent random variables under sub-linear expectation

Zhong

2017

J Inequal Appl

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“…另一方面, 文献 [60] 通过非线性偏微分积分方程定量分析了非线性 Lévy 分布, 据此, Bayraktar 和 Munk [8] 研究了非线性框架下的稳定分布和广义中心极限定理, 这些研究结果极大地促进了非线性极限理论的发展. 关于这一方面的更多研究, 参见文献 [67,78,142] 等相关文献.…”

Section: 另一方面我们的现实世界中积累了巨量的与人们的各种活动密切相关的数据但是正像前面所论述的这些数据有不可忽unclassified

Theory, methods and meaning of nonlinear expectation theory

ShiGe¹

2017

Sci. Sin.-Math.

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This is a survey on the research developments of nonlinear expectation theory. We first recall the basic definition of a space of nonlinear expectation, and then, through the representation theorem and some examples of nonlinear i.i.d. (independent and identically distributed), to explain why this new framework can be applied to calculate and quantitatively analyze probabilistic and distributional uncertainty hidden behind a real world (possibly high-dimensional) data sequence. Then we introduce two fundamentally important nonlinear normal distribution and maxima distribution and the corresponding nonlinear law of large numbers and nonlinear central limit theorem, which are crucial and fundamental breakthroughs of this new research domain. A typical application is a basic algorithm named "φ-max-mean". We also present a basic continuous-time stochastic process-nonlinear Brownian motion and its stochastic calculus, including stochastic integral, stochastic differential equations, and the corresponding nonlinear martingale theory. This new theoretical framework has generalized the axiomatical probability theory founded by Kolmogorov (1933). The key difference is the notion of nonlinear expectationÊ, whose special linear case corresponds a probability space (Ω, F, P). It is the nonlinearity that allows us to quantitatively measure the uncertainty of probabilities and probabilistic distributions inhabited in our real world. Keywords nonlinear expectation, nonlinear normal distribution, nonlinear i.i.d., nonlinear large numbers theorem and central limit theorem, nonlinear Brownian motion and its stochastic calculus, nonlinear martingale theory, nonlinear Monté-Carlo method, φ-max-mean algorithm MSC(

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“…Zhang [11] established a martingale CLT and functional CLT for sub-linear expectation under the Lindeberg condition. Wu and Chen [15] obtained a general invariance principle of G-Brownian motion for the law of the iterated logarithm for continuous bounded independent and identically distributed random variables in G-expectation space. Zhang [12] proved a new Donsker's invariance principle for independent and identically distributed random variables under the sub-linear expectation.…”

Section: §1 Introductionmentioning

confidence: 99%

Central limit theorem for linear processes generated by IID random variables under the sub-linear expectation

Liu

Zhang

2021

Appl. Math. J. Chin. Univ.

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In this paper, we investigate the central limit theorem and the invariance principle for linear processes generated by a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng [19]. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov’s central limit theorem and invariance principle to the case where probability measures are no longer additive.

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Invariance principles for the law of the iterated logarithm under G-framework

Cited by 13 publications

References 17 publications

Complete convergence and complete moment convergence for weighted sums of extended negatively dependent random variables under sub-linear expectation

Complete convergence and complete moment convergence for weighted sums of extended negatively dependent random variables under sub-linear expectation

Theory, methods and meaning of nonlinear expectation theory

Central limit theorem for linear processes generated by IID random variables under the sub-linear expectation

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