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
We give a definition of relative entropy with respect to a sublinear expectation and establish large deviation principle for the empirical measures for independent random variables under the sublinear expectation.
It is proved that the semilinear elliptic problem with zero boundary value -Δ u=λu-|u| q-1 u has a changing sign solution, as q∈(0,1) and λ>λ 2 , where λ 2 is the second eigenvalue of the ...It is proved that the semilinear elliptic problem with zero boundary value -Δ u=λu-|u| q-1 u has a changing sign solution, as q∈(0,1) and λ>λ 2 , where λ 2 is the second eigenvalue of the operator -Δ in the space H 1 0(Ω).展开更多
Sublinear expectation relaxes the linear property of classical expectation to subadditivity and positive homogeneity,which can be expressed as E(·)=sup_(θ∈θ) E_(θ)(·)for a certain set of linear expectati...Sublinear expectation relaxes the linear property of classical expectation to subadditivity and positive homogeneity,which can be expressed as E(·)=sup_(θ∈θ) E_(θ)(·)for a certain set of linear expectations{E_(θ):θ∈θ}.Such a framework can capture the uncertainty and facilitate a robust method of measuring risk loss reasonably.This study established a law of large numbers for m-dependent random vectors within the framework of sublinear expectation.Consequently,the corresponding explicit rate of convergence were derived.The results of this study can be considered as an extension of the Peng's law of large numbers[22].展开更多
In this paper,the authors firstly establish the weak laws of large numbers on the canonical space(R^(N),B(R^(N)))by traditional truncation method and Chebyshev’s inequality as in the classical probability theory.Then...In this paper,the authors firstly establish the weak laws of large numbers on the canonical space(R^(N),B(R^(N)))by traditional truncation method and Chebyshev’s inequality as in the classical probability theory.Then they extend them from the canonical space to the general sublinear expectation space.The necessary and sufficient conditions for Peng’s law of large numbers are obtained.展开更多
In this paper some oscillation criteria are established for equation [q(t)y'] +a(t)f(y) = 0, where a(t) is not assumed to be non-negative and f(y) is nodecreasing in y, and yf(y) > 0 for y≠0, f(y) also satisf...In this paper some oscillation criteria are established for equation [q(t)y'] +a(t)f(y) = 0, where a(t) is not assumed to be non-negative and f(y) is nodecreasing in y, and yf(y) > 0 for y≠0, f(y) also satisfics a sublinear condition, q(t) is a positive function on [0, ). These results extend earlier oscillation theorems of Philos and Wong.展开更多
In this paper,we develop an inexact symmetric proximal alternating direction method of multipliers(ISPADMM)with two convex combinations(ISPADMM-tcc)for solving two-block separable convex optimization problems with lin...In this paper,we develop an inexact symmetric proximal alternating direction method of multipliers(ISPADMM)with two convex combinations(ISPADMM-tcc)for solving two-block separable convex optimization problems with linear equality constraints.Specifically,the convex combination technique is incorporated into the proximal centers of both subproblems.We then approximately solve these two subproblems based on relative error criteria.The global convergence,and O(1/N)ergodic sublinear convergence rate measured by the function value residual and constraint violation are established under some mild conditions,where N denotes the number of iterations.Finally,numerical experiments on solving the l1-regularized analysis sparse recovery and the elastic net regularization regression problems illustrate the feasibility and effectiveness of the proposed method.展开更多
In this paper,we provide new sufficient conditions for the existence of positive periodic solutions for a class of indefinite singular differential equation x′′(t)+a(t)x(t)=h(t)/x^(ρ)(t)+g(t)x^(δ)(t)+e(t),whereρ...In this paper,we provide new sufficient conditions for the existence of positive periodic solutions for a class of indefinite singular differential equation x′′(t)+a(t)x(t)=h(t)/x^(ρ)(t)+g(t)x^(δ)(t)+e(t),whereρandδare two positive constants and 0<δ≤1,h,e∈L^(1)(R/TZ),g∈L^(1)(R/TZ)is positive.Our proofs are based on the fixed point theorems(Schauder’s fixed point theorem and Krasnoselskii-Guo fixed point theorem)and the positivity of the associated Green function.展开更多
This paper mainly studies the existence of positive solutions of singular sub-linear boundary value problems concerning the generalized Emden-Fowler equations. Anecessary and sufficient condition for the existence of ...This paper mainly studies the existence of positive solutions of singular sub-linear boundary value problems concerning the generalized Emden-Fowler equations. Anecessary and sufficient condition for the existence of positive solutions to this problemhas been obtained by using the method of lower and upper solutions with the fixed poilltt heorems.展开更多
In this paper, we prove that for a sublinear expectation ?[·] defined on L 2(Ω, $ \mathcal{F} $ ), the following statements are equivalent: ? is a minimal member of the set of all sublinear expectations defined ...In this paper, we prove that for a sublinear expectation ?[·] defined on L 2(Ω, $ \mathcal{F} $ ), the following statements are equivalent: ? is a minimal member of the set of all sublinear expectations defined on L 2(Ω, $ \mathcal{F} $ )? is linearthe two-dimensional Jensen’s inequality for ? holds.Furthermore, we prove a sandwich theorem for subadditive expectation and superadditive expectation.展开更多
The existence of Mather sets(generalized quasiperiodic solutions and uNlinked periodicsolutions)for sublinear Duffing equations is shown. Here the approach is based on the use ofaction-angle variables and the applicat...The existence of Mather sets(generalized quasiperiodic solutions and uNlinked periodicsolutions)for sublinear Duffing equations is shown. Here the approach is based on the use ofaction-angle variables and the application of a generalized version of Aubry-Mather theoremon semi-cylinder with finite twist assumption.展开更多
In this paper, we present some multi-dimensional central limit theorems and laws of large numbers under sublinear expectations, which extend some previous results.
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 mom...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.展开更多
In this note, we study inequality and limit theory under sublinear expectations. We mainly prove Doob＇s inequality for submartingale and Kolmogrov＇s inequality. By Kolmogrov＇s inequality, we obtain a special versio...In this note, we study inequality and limit theory under sublinear expectations. We mainly prove Doob＇s inequality for submartingale and Kolmogrov＇s inequality. By Kolmogrov＇s inequality, we obtain a special version of Kolmogrov＇s law of large numbers. Finally, we present a strong law of large numbers for independent and identically distributed random variables under one-order type moment condition.展开更多
In this note, the authors survey the existing convergence results for random variables under sublinear expectations, and prove some new results. Concretely, under the assumption that the sublinear expectation has the ...In this note, the authors survey the existing convergence results for random variables under sublinear expectations, and prove some new results. Concretely, under the assumption that the sublinear expectation has the monotone continuity property, the authors prove that convergence in capacity is stronger than convergence in distribution,and give some equivalent characterizations of convergence in distribution. In addition,they give a dominated convergence theorem under sublinear expectations, which may have its own interest.展开更多
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supported by the National Natural Science Foundation of China(11171262)the Specialized Research Fund for the Doctoral Program of Higher Education (200804860048)
文摘We give a definition of relative entropy with respect to a sublinear expectation and establish large deviation principle for the empirical measures for independent random variables under the sublinear expectation.
基金This research is supported by NNSFC(1 9771 0 72 ) and ZNSF.And thanks to JNCASR in India Fortheir host when the firstauthor is
文摘It is proved that the semilinear elliptic problem with zero boundary value -Δ u=λu-|u| q-1 u has a changing sign solution, as q∈(0,1) and λ>λ 2 , where λ 2 is the second eigenvalue of the operator -Δ in the space H 1 0(Ω).
基金funded by the National Nature Science Foundation of China(Grant No.12001128)the GuangDong Basic and Applied Basic Research Foundation(Grant No.2022A1515011899).
文摘Sublinear expectation relaxes the linear property of classical expectation to subadditivity and positive homogeneity,which can be expressed as E(·)=sup_(θ∈θ) E_(θ)(·)for a certain set of linear expectations{E_(θ):θ∈θ}.Such a framework can capture the uncertainty and facilitate a robust method of measuring risk loss reasonably.This study established a law of large numbers for m-dependent random vectors within the framework of sublinear expectation.Consequently,the corresponding explicit rate of convergence were derived.The results of this study can be considered as an extension of the Peng's law of large numbers[22].
基金supported by the National Natural Science Foundation of China(Nos.12326603,11601281,11501325)the National Key R&D Program of China(No.2018YFA0703900)+2 种基金the Natural Science Foundation of Shandong Province(No.ZR2021MA018)the Social Science Planning Project of Shandong Province(No.24CJJJ18)the Qilu Scholars Program of Shandong University。
文摘In this paper,the authors firstly establish the weak laws of large numbers on the canonical space(R^(N),B(R^(N)))by traditional truncation method and Chebyshev’s inequality as in the classical probability theory.Then they extend them from the canonical space to the general sublinear expectation space.The necessary and sufficient conditions for Peng’s law of large numbers are obtained.
文摘In this paper some oscillation criteria are established for equation [q(t)y'] +a(t)f(y) = 0, where a(t) is not assumed to be non-negative and f(y) is nodecreasing in y, and yf(y) > 0 for y≠0, f(y) also satisfics a sublinear condition, q(t) is a positive function on [0, ). These results extend earlier oscillation theorems of Philos and Wong.
基金supported by the National Natural Science Foundation of China(12171106)the Guangxi Science and Technology Program(AD23023001)+4 种基金the Natural Science Foundation of Guangxi Province(2023GXNSFBA026029)the National Natural Science Foundation of China(12401403,12361063)the Research Project of Guangxi Minzu University(2022KJQD03)the Middle-aged and Young Teachers’Basic Ability Promotion Project of Guangxi Province(2023KY0168)the Xiangsihu Young Scholars Innovative Research Team of Guangxi Minzu University(2022GXUNXSHQN04).
文摘In this paper,we develop an inexact symmetric proximal alternating direction method of multipliers(ISPADMM)with two convex combinations(ISPADMM-tcc)for solving two-block separable convex optimization problems with linear equality constraints.Specifically,the convex combination technique is incorporated into the proximal centers of both subproblems.We then approximately solve these two subproblems based on relative error criteria.The global convergence,and O(1/N)ergodic sublinear convergence rate measured by the function value residual and constraint violation are established under some mild conditions,where N denotes the number of iterations.Finally,numerical experiments on solving the l1-regularized analysis sparse recovery and the elastic net regularization regression problems illustrate the feasibility and effectiveness of the proposed method.
基金supported by the Technological Innovation Talents in Universities and Colleges in Henan Province(No.21HASTIT025)the Natural Science Foundation of Henan Province(No.222300420449)the Innovative Research Team of Henan Polytechnic University(No.T2022-7)。
文摘In this paper,we provide new sufficient conditions for the existence of positive periodic solutions for a class of indefinite singular differential equation x′′(t)+a(t)x(t)=h(t)/x^(ρ)(t)+g(t)x^(δ)(t)+e(t),whereρandδare two positive constants and 0<δ≤1,h,e∈L^(1)(R/TZ),g∈L^(1)(R/TZ)is positive.Our proofs are based on the fixed point theorems(Schauder’s fixed point theorem and Krasnoselskii-Guo fixed point theorem)and the positivity of the associated Green function.
基金supported by National Natural Science Foundation of China (Grant No.10771122)Natural Science Foundation of Shandong Province of China (Grant No.Y2006A08)National Basic Research Program of China (Grant No.2007CB814900)
文摘Under some weaker conditions,we give a central limit theorem under sublinear expectations,which extends Peng's central limit theorem.
文摘This paper mainly studies the existence of positive solutions of singular sub-linear boundary value problems concerning the generalized Emden-Fowler equations. Anecessary and sufficient condition for the existence of positive solutions to this problemhas been obtained by using the method of lower and upper solutions with the fixed poilltt heorems.
基金supported by National Basic Research Program of China (973 Program) (Grant No.2007CB814901) (Financial Risk)National Natural Science Foundation of China (Grant No. 10671111)
文摘In this paper, we prove that for a sublinear expectation ?[·] defined on L 2(Ω, $ \mathcal{F} $ ), the following statements are equivalent: ? is a minimal member of the set of all sublinear expectations defined on L 2(Ω, $ \mathcal{F} $ )? is linearthe two-dimensional Jensen’s inequality for ? holds.Furthermore, we prove a sandwich theorem for subadditive expectation and superadditive expectation.
文摘The existence of Mather sets(generalized quasiperiodic solutions and uNlinked periodicsolutions)for sublinear Duffing equations is shown. Here the approach is based on the use ofaction-angle variables and the application of a generalized version of Aubry-Mather theoremon semi-cylinder with finite twist assumption.
基金Supported by NNSFC(Grant No.11371191)Jiangsu Province Basic Research Program(Natural Science Foundation)(Grant No.BK2012720)
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.
基金supported in part by National Basic Research Program of China (973 Program) (Grant No. 2007CB814901)the Natural Science Foundation of Shandong Province (Grant No. ZR2009AL015)
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.
基金Supported in part by the National Natural Science Foundation of China under Grant No.11371191Jiangsu Province Basic Research Program(Natural Science Foundation)under Grant No.BK2012720
文摘In this note, we study inequality and limit theory under sublinear expectations. We mainly prove Doob＇s inequality for submartingale and Kolmogrov＇s inequality. By Kolmogrov＇s inequality, we obtain a special version of Kolmogrov＇s law of large numbers. Finally, we present a strong law of large numbers for independent and identically distributed random variables under one-order type moment condition.
基金supported by the National Natural Science Foundation of China(No.11771309)the Fundamental Research Funds for the Central Universities of China
文摘In this note, the authors survey the existing convergence results for random variables under sublinear expectations, and prove some new results. Concretely, under the assumption that the sublinear expectation has the monotone continuity property, the authors prove that convergence in capacity is stronger than convergence in distribution,and give some equivalent characterizations of convergence in distribution. In addition,they give a dominated convergence theorem under sublinear expectations, which may have its own interest.
