In this article,we establish a general result on complete moment convergence for arrays of rowwise negatively dependent(ND)random variables under the sub-linear expectations.As applications,we can obtain a series of r...In this article,we establish a general result on complete moment convergence for arrays of rowwise negatively dependent(ND)random variables under the sub-linear expectations.As applications,we can obtain a series of results on complete moment convergence for ND random variables under the sub-linear expectations.展开更多
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...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.展开更多
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 expec...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.展开更多
In this article,we study strong limit theorems for weighted sums of extended negatively dependent random variables under the sub-linear expectations.We establish general strong law and complete convergence theorems fo...In this article,we study strong limit theorems for weighted sums of extended negatively dependent random variables under the sub-linear expectations.We establish general strong law and complete convergence theorems for weighted sums of extended negatively dependent random variables under the sub-linear expectations.Our results of strong limit theorems are more general than some related results previously obtained by Thrum(1987),Li et al.(1995)and Wu(2010)in classical probability space.展开更多
In this paper, by establishing a Borel–Cantelli lemma for a capacity which is not necessarily continuous, and a link between a sequence of independent random variables under the sub-linear expectation and a sequence ...In this paper, by establishing a Borel–Cantelli lemma for a capacity which is not necessarily continuous, and a link between a sequence of independent random variables under the sub-linear expectation and a sequence of independent random variables on R^(∞) under a probability, we give the sufficient and necessary conditions of the strong law of large numbers for independent and identically distributed random variables under the sub-linear expectation, and the sufficient and necessary conditions for the convergence of an infinite series of independent random variables, without the assumption on the continuity of the capacities. A purely probabilistic proof of a weak law of large numbers is also given.展开更多
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 ...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.展开更多
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 logari...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.展开更多
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 distr...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.展开更多
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 indepen...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.展开更多
Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper est...Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper establishes the Kolmogorov type exponential inequalities of the partial sums of independent random variables as well as negatively dependent random variables under the sub-linear expectations. As applications of the exponential inequalities, the laws of the iterated logarithm in the sense of non-additive capacities are proved for independent or negatively dependent identically distributed random variables with finite second order moments.For deriving a lower bound of an exponential inequality, a central limit theorem is also proved under the sublinear expectation for random variables with only finite variances.展开更多
The central limit theorem of martingales is the fundamental tool for studying the convergence of stochastic processes,especially stochastic integrals and differential equations.In this paper,the central limit theorem ...The central limit theorem of martingales is the fundamental tool for studying the convergence of stochastic processes,especially stochastic integrals and differential equations.In this paper,the central limit theorem and the functional central limit theorem are obtained for martingale-like random variables under the sub-linear expectation.As applications,the Lindeberg's central limit theorem is obtained for independent but not necessarily identically distributed random variables,and a new proof of the Lévy characterization of a GBrownian motion without using stochastic calculus is given.For proving the results,Rosenthal's inequality and the exponential inequality for the martingale-like random variables are established.展开更多
In this paper,we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space,where the random variables are not necessarily identically distribu...In this paper,we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space,where the random variables are not necessarily identically distributed.Exponential inequalities for the maximum sum of independent random variables and Kolmogorov’s converse exponential inequalities are established as tools for showing the law of the iterated logarithm.As an application,the sufficient and necessary conditions of the law of the iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained.In the paper,it is also shown that if the sub-linear expectation space is rich enough,it will have no continuous capacity.The laws of the iterated logarithm are established without the assumption on the continuity of capacities.展开更多
Under the framework of sub-linear expectation initiated by Peng,motivated by the concept of extended negative dependence,we establish a law of logarithm for arrays of row-wise extended negatively dependent random vari...Under the framework of sub-linear expectation initiated by Peng,motivated by the concept of extended negative dependence,we establish a law of logarithm for arrays of row-wise extended negatively dependent random variables under weak conditions.Besides,the law of logarithm for independent and identically distributed arrays is derived more precisely and the sufficient and necessary conditions for the law of logarithm are obtained.展开更多
In this paper,some laws of large numbers are established for random variables that satisfy the Pareto distribution,so that the relevant conclusions in the traditional probability space are extended to the sub-linear e...In this paper,some laws of large numbers are established for random variables that satisfy the Pareto distribution,so that the relevant conclusions in the traditional probability space are extended to the sub-linear expectation space.Based on the Pareto distribution,we obtain the weak law of large numbers and strong law of large numbers of the weighted sum of some independent random variable sequences.展开更多
In this note,we establish a compact law of the iterated logarithm under the upper capacity for independent and identically distributed random variables in a sub-linear expectation space.For showing the result,a self-n...In this note,we establish a compact law of the iterated logarithm under the upper capacity for independent and identically distributed random variables in a sub-linear expectation space.For showing the result,a self-normalized law of the iterated logarithm is established.展开更多
基金the National Natural Science Foundation of China(71871046,11661029)Natural Science Foundation of Guangxi(2018JJB110010)。
文摘In this article,we establish a general result on complete moment convergence for arrays of rowwise negatively dependent(ND)random variables under the sub-linear expectations.As applications,we can obtain a series of results on complete moment convergence for ND random variables under the sub-linear expectations.
基金Research supported by grants from the NSF of China(1173101212031005)+2 种基金Ten Thousands Talents Plan of Zhejiang Province(2018R52042)NSF of Zhejiang Province(LZ21A010002)the Fundamental Research Funds for the Central Universities。
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.
基金Supported by the National Natural Science Foundation of China(11771178)the Science and Technology Development Program of Jilin Province(20170101152JC)+1 种基金the Science and Technology Program of Jilin Edu-cational Department during the“13th Five-Year”Plan Period(JJKH20200951KJ)Fundamental Research Funds for the Central Universities。
文摘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.
基金supported by the Natural Science Foundation of Guangxi(Grant No.2024GXNSFAA010476)the National Natural Science Foundation of China(Grant No.12361031)。
文摘In this article,we study strong limit theorems for weighted sums of extended negatively dependent random variables under the sub-linear expectations.We establish general strong law and complete convergence theorems for weighted sums of extended negatively dependent random variables under the sub-linear expectations.Our results of strong limit theorems are more general than some related results previously obtained by Thrum(1987),Li et al.(1995)and Wu(2010)in classical probability space.
基金Supported by grants from the NSF of China(Grant Nos.11731012,12031005)Ten Thousands Talents Plan of Zhejiang Province(Grant No.2018R52042)+1 种基金NSF of Zhejiang Province(Grant No.LZ21A010002)the Fundamental Research Funds for the Central Universities。
文摘In this paper, by establishing a Borel–Cantelli lemma for a capacity which is not necessarily continuous, and a link between a sequence of independent random variables under the sub-linear expectation and a sequence of independent random variables on R^(∞) under a probability, we give the sufficient and necessary conditions of the strong law of large numbers for independent and identically distributed random variables under the sub-linear expectation, and the sufficient and necessary conditions for the convergence of an infinite series of independent random variables, without the assumption on the continuity of the capacities. A purely probabilistic proof of a weak law of large numbers is also given.
基金Supported by the NSF of China(Grant No.11731012)the 973 Program(Grant No.2015CB352302)+1 种基金Zhejiang Provincial Natural Science Foundation(Grant No.LY17A010016)the Fundamental Research Funds for the Central Universities
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.
基金This research supported by Grants from the National Natural Science Foundation of China(No.11225104)and the Fundamental Research Funds for the Central Universities.
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.
基金Supported by grants from the NSF of China(Grant No.11731012)Ten Thousands Talents Plan of Zhejiang Province(Grant No.2018R52042)+1 种基金the 973 Program(Grant No.2015CB352302)the Fundamental Research Funds for the Central Universities。
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.
基金supported by National Natural Science Foundation of China(Grant No.11225104)the Fundamental Research Funds for the Central Universities
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.
基金supported by National Natural Science Foundation of China (Grant No. 11225104)the National Basic Research Program of China (Grant No. 2015CB352302)the Fundamental Research Funds for the Central Universities
文摘Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper establishes the Kolmogorov type exponential inequalities of the partial sums of independent random variables as well as negatively dependent random variables under the sub-linear expectations. As applications of the exponential inequalities, the laws of the iterated logarithm in the sense of non-additive capacities are proved for independent or negatively dependent identically distributed random variables with finite second order moments.For deriving a lower bound of an exponential inequality, a central limit theorem is also proved under the sublinear expectation for random variables with only finite variances.
基金supported by National Natural Science Foundation of China(Grant No.11731012)the Fundamental Research Funds for the Central Universities+1 种基金the State Key Development Program for Basic Research of China(Grant No.2015CB352302)Zhejiang Provincial Natural Science Foundation(Grant No.LY17A010016)。
文摘The central limit theorem of martingales is the fundamental tool for studying the convergence of stochastic processes,especially stochastic integrals and differential equations.In this paper,the central limit theorem and the functional central limit theorem are obtained for martingale-like random variables under the sub-linear expectation.As applications,the Lindeberg's central limit theorem is obtained for independent but not necessarily identically distributed random variables,and a new proof of the Lévy characterization of a GBrownian motion without using stochastic calculus is given.For proving the results,Rosenthal's inequality and the exponential inequality for the martingale-like random variables are established.
基金the National Natural Science Foundation of China(Grant Nos.11731012,12031005)Ten Thousand Talents Plan of Zhejiang Province(Grant No.2018R52042)+1 种基金Natural Science Foundation of Zhejiang Province(Grant No.LZ21A010002)the Fundamental Research Funds for the Central Universities.
文摘In this paper,we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space,where the random variables are not necessarily identically distributed.Exponential inequalities for the maximum sum of independent random variables and Kolmogorov’s converse exponential inequalities are established as tools for showing the law of the iterated logarithm.As an application,the sufficient and necessary conditions of the law of the iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained.In the paper,it is also shown that if the sub-linear expectation space is rich enough,it will have no continuous capacity.The laws of the iterated logarithm are established without the assumption on the continuity of capacities.
基金supported by grants from the National Natural Science Foundation of China (No. 11731012)Ten Thousands Talents Plan of Zhejiang Province (Grant No. 2018R52042)the Fundamental Research Funds for the Central Universities
文摘Under the framework of sub-linear expectation initiated by Peng,motivated by the concept of extended negative dependence,we establish a law of logarithm for arrays of row-wise extended negatively dependent random variables under weak conditions.Besides,the law of logarithm for independent and identically distributed arrays is derived more precisely and the sufficient and necessary conditions for the law of logarithm are obtained.
基金AcknowledgmentssThe authors thank the National Natural Science Foundation of China(Grant No.12061028)Guangxi Natural Science Foundation Joint Incubation Project(Grant No.2018GXNSFAA294131)+1 种基金Guangxi Natural Science Foundation(Grant No.2018G XNSFAA281011)Innovation Project of Guangxi Graduate Education(Grant No.YCSW2020175)for their financial support。
文摘In this paper,some laws of large numbers are established for random variables that satisfy the Pareto distribution,so that the relevant conclusions in the traditional probability space are extended to the sub-linear expectation space.Based on the Pareto distribution,we obtain the weak law of large numbers and strong law of large numbers of the weighted sum of some independent random variable sequences.
文摘In this note,we establish a compact law of the iterated logarithm under the upper capacity for independent and identically distributed random variables in a sub-linear expectation space.For showing the result,a self-normalized law of the iterated logarithm is established.
