Assume that{a_(i),−∞<i<∞}is an absolutely summable sequence of real numbers.We establish the complete q-order moment convergence for the partial sums of moving average processes{X_(n)=Σ_(i=−∞)^(∞)a_(i)Y_(i+...Assume that{a_(i),−∞<i<∞}is an absolutely summable sequence of real numbers.We establish the complete q-order moment convergence for the partial sums of moving average processes{X_(n)=Σ_(i=−∞)^(∞)a_(i)Y_(i+n),n≥1}under some proper conditions,where{Yi,-∞<i<∞}is a doubly infinite sequence of negatively dependent random variables under sub-linear expectations.These results extend and complement the relevant results in probability space.展开更多
In this work, the sample path large deviations for independent, identically distributed random variables under sub-linear expectations are established. The results obtained in sublinear expectation spaces extend the c...In this work, the sample path large deviations for independent, identically distributed random variables under sub-linear expectations are established. The results obtained in sublinear expectation spaces extend the corresponding ones in probability space.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 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
We establish the Strassen's law of the iterated logarithm(LIL for short)for independent and identically distributed random variables with E[X_(1)]=ε[X_(1)]=0 and Cv[X_(1)^(2)]<∞ under a sublinear expectation ...We establish the Strassen's law of the iterated logarithm(LIL for short)for independent and identically distributed random variables with E[X_(1)]=ε[X_(1)]=0 and Cv[X_(1)^(2)]<∞ under a sublinear expectation space with a countably sub-additive capacity V.We also show the LIL for upper capacity with σ=σ under some certain conditions.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
The complete convergence for weighted sums of sequences of independent,identically distributed random variables under sublinear expectation space is studied.By moment inequality and truncation methods,we establish the...The complete convergence for weighted sums of sequences of independent,identically distributed random variables under sublinear expectation space is studied.By moment inequality and truncation methods,we establish the equivalent conditions of complete convergence for weighted sums of sequences of independent,identically distributed random variables under sublinear expectation space.The results complement the corresponding results in probability space to those for sequences of independent,identically distributed random variables under sublinear expectation space.展开更多
Employees’innovative performance determines an organization’s innovation,which critically impacts its structural optimization and sustainability.Applying expectancy theory,we examined how and when the Pygmalion effe...Employees’innovative performance determines an organization’s innovation,which critically impacts its structural optimization and sustainability.Applying expectancy theory,we examined how and when the Pygmalion effect occurs in the relationship between leaders’expectations of innovation and employee innovative behavior.Our sample comprised 201 frontline employees(female=31.84%;mean age=41.48 years,SD=7.97 years)in a Chinese coal enterprise,who completed surveys on innovation expectations of leaders,expected positive performance outcomes,innovative self-efficacy and innovative behavior.The results revealed that employees’expected positive performance outcomes mediated the positive relationship between leaders’innovation expectations and their innovative behavior.Employees’innovative self-efficacy positively moderated the relationship between their expected positive performance outcomes and innovative behavior,with this relationship being stronger for employees with high innovative self-efficacy.Moreover,we validated the moderated mediation model.Findings suggest that leaders can stimulate employee innovative behavior through expressing expectations and they also need to consider the boundary conditions.展开更多
Objective:The aim of this study was to investigate the health expectations(HEs)in patients with diabetes and the factors affecting it using an integrative systematic review.Methods:A comprehensive literature search wa...Objective:The aim of this study was to investigate the health expectations(HEs)in patients with diabetes and the factors affecting it using an integrative systematic review.Methods:A comprehensive literature search was conducted in databases including PubMed,Scopus,Science Direct,Google Scholar,ProQuest,CINAHL,SID,Iran Doc,and Magiran without time limitation using related keywords.The systematic literature search was performed using a guide provided by York University.Results:From a total of 1883 articles,17 of them finally met the eligibility criteria.According to the findings of this study,HE of diabetic patients was included in four categories:“dynamism,”“consequentialism,”“comprehensive care,”and“access to facilities.”Also,these expectations can be influenced by various factors,including“developments in societies,”“changing the conditions of health care systems,”and“individual factors.”Conclusions:Health professionals should pay attention to the expectations of diabetic patients who are involved in complex disease conditions and consider meeting them as a key factor in providing patient-centered care,improving health outcomes,and patient satisfaction.展开更多
Objective:This study aimed to investigate the learning expectations of undergraduate nursing students regarding the flipped Health Assessment course.Methods:This descriptive,qualitative study was conducted at a medica...Objective:This study aimed to investigate the learning expectations of undergraduate nursing students regarding the flipped Health Assessment course.Methods:This descriptive,qualitative study was conducted at a medical university in Fuzhou,Fujian Province,China.An interview outline was designed based on the core dimensions of Expectation Confirmation Theory(expectation sources,expectation content,and expectation importance).Thirty second-year undergraduate nursing students who had completed first-year basic medical courses and were about to take the flipped Health Assessment course were interviewed between June and July 2022.Interview data were analyzed using qualitative content analysis.Results:Five major themes and thirteen subthemes were identified.Theme 1 was expectation sourcesperceived learning difficulties from past experiences,which included four subthemes:insufficient autonomous learning ability,confusion regarding learning methods,insufficient engagement in learning,and low professional identity.Theme 2 included knowledge and information expectations,which comprised three subthemes:knowledge to improve professional competence,knowledge to enhance academic competitiveness,and knowledge to boost self-efficacy.Theme 3 comprised logical expectations and included two subthemes:flexible teaching methods and efficient instructional tools.Theme 4,pleasure expectations,included two subthemes:vivid teaching styles and diversified teaching evaluations.Theme 5 comprised professional value expectations and included two subthemes:teachers'responsible professional attitudes and gentle emotional support.Conclusion:Students'learning expectations originate from perceived learning difficulties,such as insufficient learning autonomy and engagement,confusion about learning methods,and a lack of professional identity.They showed various learning expectations for the flipped Health Assessment course,including knowledge and information,logic,pleasure,and professional value expectations.展开更多
This study investigated the relationship between parental cognitive ability and child logical reasoning ability,and the role of academic expectation and family environment in that relationship.Based on the 2020 China ...This study investigated the relationship between parental cognitive ability and child logical reasoning ability,and the role of academic expectation and family environment in that relationship.Based on the 2020 China Family Panel Studies(CFPS)data,1491 children(girls ratio=53.78%;average grade=6.023 years,school grade standard deviation=1.825 years).Results following multiple regression model(OLS)show that the higher the parental cognitive ability,the higher the children’s logical reasoning ability.Secondly,parental academic expectation serves as a mediator between their cognitive ability and children’s logical reasoning ability for higher logical reasoning by children.Third,a possible family environment acts as a mediator in the relationship between parents’cognitive ability and children’s logical reasoning ability to be higher.We conclude from thesefindings that parents with high cognitive abilities can enhance their children’s logical reasoning skills not only by setting higher academic expectations,but also by cultivating a supportive family environment.Thesefindings imply a need for intervention to improve family quality of life to enhance children’s thinking abilities to optimize their academic learning.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
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
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
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, 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,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
基金Supported by the Academic Achievement Re-cultivation Projects of Jingdezhen Ceramic University(Grant Nos.215/20506341215/20506277)the Doctoral Scientific Research Starting Foundation of Jingdezhen Ceramic University(Grant No.102/01003002031)。
文摘Assume that{a_(i),−∞<i<∞}is an absolutely summable sequence of real numbers.We establish the complete q-order moment convergence for the partial sums of moving average processes{X_(n)=Σ_(i=−∞)^(∞)a_(i)Y_(i+n),n≥1}under some proper conditions,where{Yi,-∞<i<∞}is a doubly infinite sequence of negatively dependent random variables under sub-linear expectations.These results extend and complement the relevant results in probability space.
基金Doctoral Scientific Research Starting Foundation of Jingdezhen Ceramic University (Grant No. 102/01003002031)Academic Achievement Re-cultivation Project of Jingdezhen Ceramic University (Grant No. 215/205062777)the Science and Technology Research Project of Jiangxi Provincial Department of Education of China (Grant No. GJJ2201041)。
文摘In this work, the sample path large deviations for independent, identically distributed random variables under sub-linear expectations are established. The results obtained in sublinear expectation spaces extend the corresponding ones in probability space.
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Supported by grants from the NSF of China(Grant Nos.2024YFA101350,U23A2064 and 12031005)。
文摘We establish the Strassen's law of the iterated logarithm(LIL for short)for independent and identically distributed random variables with E[X_(1)]=ε[X_(1)]=0 and Cv[X_(1)^(2)]<∞ under a sublinear expectation space with a countably sub-additive capacity V.We also show the LIL for upper capacity with σ=σ under some certain conditions.
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 Doctoral Scientific Research Starting Foundation of Jingdezhen Ceramic University(Grant No.102/01003002031)Re-accompanying Funding Project of Academic Achievements of Jingdezhen Ceramic University(Grant Nos.215/20506277,215/20506341)。
文摘The complete convergence for weighted sums of sequences of independent,identically distributed random variables under sublinear expectation space is studied.By moment inequality and truncation methods,we establish the equivalent conditions of complete convergence for weighted sums of sequences of independent,identically distributed random variables under sublinear expectation space.The results complement the corresponding results in probability space to those for sequences of independent,identically distributed random variables under sublinear expectation space.
基金funded by“National Natural Science Foundation of China”grant number[71872119]“Shanxi Provincial Government Major Decision-making Consulting Project”grant number[ZB20240114]The APC was funded by“Shanxi Provincial Government Major Decision-making Consulting Project”grant number[ZB20240114].
文摘Employees’innovative performance determines an organization’s innovation,which critically impacts its structural optimization and sustainability.Applying expectancy theory,we examined how and when the Pygmalion effect occurs in the relationship between leaders’expectations of innovation and employee innovative behavior.Our sample comprised 201 frontline employees(female=31.84%;mean age=41.48 years,SD=7.97 years)in a Chinese coal enterprise,who completed surveys on innovation expectations of leaders,expected positive performance outcomes,innovative self-efficacy and innovative behavior.The results revealed that employees’expected positive performance outcomes mediated the positive relationship between leaders’innovation expectations and their innovative behavior.Employees’innovative self-efficacy positively moderated the relationship between their expected positive performance outcomes and innovative behavior,with this relationship being stronger for employees with high innovative self-efficacy.Moreover,we validated the moderated mediation model.Findings suggest that leaders can stimulate employee innovative behavior through expressing expectations and they also need to consider the boundary conditions.
基金supported by Tehran University of Medical Sciences,Tehran,Iran(no.9711199007).
文摘Objective:The aim of this study was to investigate the health expectations(HEs)in patients with diabetes and the factors affecting it using an integrative systematic review.Methods:A comprehensive literature search was conducted in databases including PubMed,Scopus,Science Direct,Google Scholar,ProQuest,CINAHL,SID,Iran Doc,and Magiran without time limitation using related keywords.The systematic literature search was performed using a guide provided by York University.Results:From a total of 1883 articles,17 of them finally met the eligibility criteria.According to the findings of this study,HE of diabetic patients was included in four categories:“dynamism,”“consequentialism,”“comprehensive care,”and“access to facilities.”Also,these expectations can be influenced by various factors,including“developments in societies,”“changing the conditions of health care systems,”and“individual factors.”Conclusions:Health professionals should pay attention to the expectations of diabetic patients who are involved in complex disease conditions and consider meeting them as a key factor in providing patient-centered care,improving health outcomes,and patient satisfaction.
基金This research received grant from 2022 Undergraduate Education and Teaching Research Project of Fujian Medical University(22028).
文摘Objective:This study aimed to investigate the learning expectations of undergraduate nursing students regarding the flipped Health Assessment course.Methods:This descriptive,qualitative study was conducted at a medical university in Fuzhou,Fujian Province,China.An interview outline was designed based on the core dimensions of Expectation Confirmation Theory(expectation sources,expectation content,and expectation importance).Thirty second-year undergraduate nursing students who had completed first-year basic medical courses and were about to take the flipped Health Assessment course were interviewed between June and July 2022.Interview data were analyzed using qualitative content analysis.Results:Five major themes and thirteen subthemes were identified.Theme 1 was expectation sourcesperceived learning difficulties from past experiences,which included four subthemes:insufficient autonomous learning ability,confusion regarding learning methods,insufficient engagement in learning,and low professional identity.Theme 2 included knowledge and information expectations,which comprised three subthemes:knowledge to improve professional competence,knowledge to enhance academic competitiveness,and knowledge to boost self-efficacy.Theme 3 comprised logical expectations and included two subthemes:flexible teaching methods and efficient instructional tools.Theme 4,pleasure expectations,included two subthemes:vivid teaching styles and diversified teaching evaluations.Theme 5 comprised professional value expectations and included two subthemes:teachers'responsible professional attitudes and gentle emotional support.Conclusion:Students'learning expectations originate from perceived learning difficulties,such as insufficient learning autonomy and engagement,confusion about learning methods,and a lack of professional identity.They showed various learning expectations for the flipped Health Assessment course,including knowledge and information,logic,pleasure,and professional value expectations.
基金supported by scientific research fund of Jiangxi Provincial Social Sciences“14th Five-Year Plan”(No.23SH05).
文摘This study investigated the relationship between parental cognitive ability and child logical reasoning ability,and the role of academic expectation and family environment in that relationship.Based on the 2020 China Family Panel Studies(CFPS)data,1491 children(girls ratio=53.78%;average grade=6.023 years,school grade standard deviation=1.825 years).Results following multiple regression model(OLS)show that the higher the parental cognitive ability,the higher the children’s logical reasoning ability.Secondly,parental academic expectation serves as a mediator between their cognitive ability and children’s logical reasoning ability for higher logical reasoning by children.Third,a possible family environment acts as a mediator in the relationship between parents’cognitive ability and children’s logical reasoning ability to be higher.We conclude from thesefindings that parents with high cognitive abilities can enhance their children’s logical reasoning skills not only by setting higher academic expectations,but also by cultivating a supportive family environment.Thesefindings imply a need for intervention to improve family quality of life to enhance children’s thinking abilities to optimize their academic learning.
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 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.
