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| 1 | Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications显示文摘Classical Kolmogorov's and Rosenthal's inequalities for the maximum partial sums of random variables are basic tools for studying the strong laws of large numbers.In this paper,motived by the notion of independent and identically distributed random variables under the sub-linear expectation initiated by Peng(2008),we introduce the concept of negative dependence of random variables and establish Kolmogorov's and Rosenthal's inequalities for the maximum partial sums of negatively dependent random variables under the sub-linear expectations.As an application,we show that Kolmogorov's strong law of larger numbers holds for independent and identically distributed random variables under a continuous sub-linear expectation if and only if the corresponding Choquet integral is finite. | ZHANG LiXin | 2016 | Science China Mathematics2016，59，4： | 46 |
| 2 | Strong laws of large numbers for sub-linear expectations显示文摘We investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed(IID) random variables for sub-linear expectations initiated by Peng.It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov's strong law of large numbers to the case where probability measures are no longer additive. An important feature of these strong laws of large numbers is to provide a frequentist perspective on capacities. | CHEN ZengJing | 2016 | Science China Mathematics2016，59，5： | 25 |
| 3 | 非线性期望的理论、方法及意义显示文摘本文是非线性期望理论进展的一个综述,首先给出非线性期望空间的基本定义,并通过非线性期望的表示定理和几个典型的非线性独立同分布(i.i.d.)的例子来说明为什么这个新框架可以广泛地用来分析和计算现实世界(高维)数据背后隐藏的概率和统计分布的不确定性;进而介绍次线性期望空间中两个最重要的统计分布—非线性正态分布和最大分布,以及相应的非线性大数定律和中心极限定理,是新领域的基础性和关键性的突破,其典型的应用就是对于现实的(高维)样本数据的非常简单而深刻的φ-max-mean算法.本文还介绍一个最重要的连续时间随机过程——非线性Brown运动及相关随机分析,包括随机积分、随机微分方程和非线性鞅理论.新的理论框架实质性地推广了Kolmogorov于1933年建立的、以概率测度为核心的概率论公理体系(?,F,P).其关键不同的是,其核心概念是(非线性)期望ê,期望为线性的特殊情形对应着概率论公理体系.正是这种非线性使人们能够对于现实世界中无处不在的概率模型本身的不确定性也能进行定量的分析和计算.从而实质性地放宽了概率统计理论中对于现实世界的随机数据的统计假设要求,本文也因而获得了实际样本数据的非线性分布的φ-max-mean算法,它是一种新的非线性Monté-Carlo算法. | 彭实戈 | 2017 | 中国科学:数学2017，47，10： | 18 |
| 4 | Donsker’s Invariance Principle Under the Sub-linear Expectation with an Application to Chung’s Law of the Iterated Logarithm显示文摘We prove a new Donsker’s invariance principle for independent and identically distributed random variables under the sub-linear expectation.As applications,the small deviations and Chung’s law of the iterated logarithm are obtained. | Li-Xin Zhang | 2015 | Communications in Mathematics and Statistics2015，3，2： | 17 |
| 5 | A general central limit theorem under sublinear expectations显示文摘Under some weaker conditions,we give a central limit theorem under sublinear expectations,which extends Peng's central limit theorem. | LI Min & SHI YuFeng School of Mathematics,Shandong University,Jinan 250100,China | 2010 | Science China Mathematics2010，53，8： | 15 |
| 6 | Invariance principles for the law of the iterated logarithm under G-framework显示文摘We obtain a general invariance principle of G-Brownian motion for the law of the iterated logarithm(LIL for short). For continuous bounded independent and identically distributed random variables in G-expectation space, we also give an invariance principle for LIL. In some sense, this result is an extension of the classical Strassen's invariance principle to the case where probability measure is no longer additive. Furthermore,we give some examples as applications. | WU PanYu CHEN ZengJing | 2015 | Science China Mathematics2015，58，6： | 7 |
| 7 | Three Series Theorem for Independent Random Variables under Sub-linear Expectations with Applications显示文摘In this paper, motived by the notion of independent and identically distributed random variables under the sub-linear expectation initiated by Peng, we establish a three series theorem of independent random variables under the sub-linear expectations. As an application, we obtain the Marcinkiewicz's strong law of large numbers for independent and identically distributed random variables under the sub-linear expectations. The technical details are different from those for classical theorems because the sub-linear expectation and its related capacity are not additive. | Jia Pan XU Li Xin ZHANG | 2019 | Acta Mathematica Sinica,English Series2019，35，2： | 6 |
| 8 | Moment bounds for IID sequences under sublinear expectations显示文摘With the notion of independent identically distributed(IID) random variables under sublinear expectations introduced by Peng,we investigate moment bounds for IID sequences under sublinear expectations. We obtain a moment inequality for a sequence of IID random variables under sublinear expectations. As an application of this inequality,we get the following result:For any continuous functionsatisfying the growth condition |(x) | C(1 + |x|p) for some C > 0,p 1 depending on ,the central limit theorem under sublinear expectations obtained by Peng still holds. | HU Feng1,2 1Department of Mathematics,Qufu Normal University,Qufu 273165,China 2School of Mathematics,Shandong University,Jinan 250100,China | 2011 | Science China Mathematics2011，54，10： | 6 |
| 9 | The Convergence of the Sums of Independent Random Variables Under the Sub-linear Expectations显示文摘Let {Xn;n≥1} be a sequence of independent random variables on a probability space(Ω,F,P) and Sn=∑k=1n Xk.It is well-known that the almost sure convergence,the convergence in probability and the convergence in distribution of Sn are equivalent.In this paper,we prove similar results for the independent random variables under the sub-linear expectations,and give a group of sufficient and necessary conditions for these convergence.For proving the results,the Levy and Kolmogorov maximal inequalities for independent random variables under the sub-linear expectation are established.As an application of the maximal inequalities,the sufficient and necessary conditions for the central limit theorem of independent and identically distributed random variables are also obtained. | Li Xin ZHANG | 2020 | Acta Mathematica Sinica,English Series2020，36，3： | 5 |
| 10 | STRONG LIMIT THEOREMS FOR EXTENDED INDEPENDENT RANDOM VARIABLES AND EXTENDED NEGATIVELY DEPENDENT RANDOM VARIABLES UNDER SUB-LINEAR EXPECTATIONS显示文摘Limit theorems for non-additive probabilities or non-linear expectations are challenging issues which have attracted a lot of interest recently.The purpose of this paper is to study the strong law of large numbers and the law of the iterated logarithm for a sequence of random variables in a sub-linear expectation space under a concept of extended independence which is much weaker and easier to verify than the independence proposed by Peng[20].We introduce a concept of extended negative dependence which is an extension of the kind of weak independence and the extended negative independence relative to classical probability that has appeared in the recent literature.Powerful tools such as moment inequality and Kolmogorov’s exponential inequality are established for these kinds of extended negatively independent random variables,and these tools improve a lot upon those of Chen,Chen and Ng[1].The strong law of large numbers and the law of iterated logarithm are also obtained by applying these inequalities. | 张立新 | 2022 | Acta Mathematica Scientia2022，42，2： | 5 |
| 11 | Peng g-期望下的大数定律显示文摘Peng于1997年通过倒向随机微分方程引入了一类性质很好的非线性数学期望,即g-期望.本文中,我们将给出Pengg-期望下的弱大数定律与强大数定律. | 林乾 石玉峰 | 2012 | 中国科学:数学2012，42，4： | 4 |
| 12 | Numerical simulations for G-Brownian motion显示文摘 | Jie YANG Weidong ZHA0 | 2016 | Frontiers of Mathematics in China2016，11，6： | 4 |
| 13 | Multi-dimensional Central Limit Theorems and Laws of Large Numbers under Sublinear Expectations显示文摘In this paper, we present some multi-dimensional central limit theorems and laws of large numbers under sublinear expectations, which extend some previous results. | Ze Chun HU Ling ZHOU | 2015 | Acta Mathematica Sinica,English Series2015，31，2： | 4 |
| 14 | Multiple G-Ito integral in G-expectation space显示文摘 | Panyu WU | 2013 | Frontiers of Mathematics in China2013，8，2： | 3 |
| 15 | How big are the increments of G-Brownian motion?显示文摘In this paper,we investigate the problem:How big are the increments of G-Brownian motion.We obtain the Csrg and R′ev′esz’s type theorem for the increments of G-Brownian motion.As applications of this result,we get the law of iterated logarithm and the Erds and R′enyi law of large numbers for G-Brownian motion.Furthermore,it turns out that our theorems are natural extensions of the classical results obtained by Csrg and R′ev′esz(1979). | HU Feng CHEN ZengJing ZHANG DeFei | 2014 | Science China Mathematics2014，57，8： | 3 |
| 16 | Differentiability of stochastic differential equations driven by the G-Brownian motion显示文摘In this paper,we study the differentiability of the solutions of stochastic differential equations driven by the G-Brownian motion with respect to the initial data and the parameter. | LIN Qian | 2013 | Science China Mathematics2013，56，5： | 3 |
| 17 | Rosenthal's Inequalities for Asymptotically Almost Negatively Associated Random Variables Under Upper Expectations显示文摘In this paper, the authors generalize the concept of asymptotically almost negatively associated random variables from the classic probability space to the upper expectation space. Within the framework, the authors prove some different types of Rosenthal's inequalities for sub-additive expectations. Finally, the authors prove a strong law of large numbers as the application of Rosenthal's inequalities. | Ning ZHANG Yuting LAN | 2019 | Chinese Annals of Mathematics,Series B2019，40，1： | 3 |
| 18 | A Worst-Case Risk Measure by G-VaR显示文摘G-VaR,which is a type of worst-case value-at-risk(VaR),is defined as measuring risk incorporating model uncertainty.Compared with most extant notions of worst-case VaR,G-VaR can be computed using an explicit formula,and can be applied to large portfolios of several hundred dimensions with low computational cost.We also apply G-VaR to robust portfolio optimization,thereby providing a tractable means to facilitate optimal allocations under the condition of market ambiguity. | Zi-ting PEI Xi-shun WANG Yu-hong XU Xing-ye YUE | 2021 | Acta Mathematicae Applicatae Sinica2021，37，2： | 2 |
| 19 | Some properties of g-convex functions显示文摘In this paper,we obtain that eachg-convex function is continuous and convex,and we also extend Jia and Peng’s result on the characterization of g-convex function without the bounded assumption of the value of g at the origin. | LI XiaoJuan | 2013 | Science China Mathematics2013，56，10： | 2 |
| 20 | Self-Normalized Moderate Deviation and Laws of the Iterated Logarithm Under G-Expectation显示文摘The sub-linear expectation or called G-expectation is a non-linear expectation having advantage of modeling non-additive probability problems and the volatilityuncertainty in finance.Let{Xn;n≥1}be a sequence of independent random vari-ables in a sub-linear expectation space(Ω,H,E^(^)).Denote S_(n)=∑_(k=1)^(n)Xk and=V_(n)^(2)=∑_(k=1)^(n)X_(k)^(2).In this paper,a moderate deviation for self-normalized sums,thatis,the asymptotic capacity of the event{Sn/Vn≥x_(n)}for x_(n)=o(√n),is found both for identically distributed random variables and independent but not necessarilyidentically distributed random variables.As an application,the self-normalized lawsof the iterated logarithm are obtained.A Bernstein's type inequality is also establishedfor proving the law of the iterated logarithm. | Li-Xin Zhang | 2016 | Communications in Mathematics and Statistics2016，4，2： | 2 |
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