Chaundy and Jolliffe proved that if {a n } is a non-increasing (monotonic) real sequence with lim n →∞ a n = 0, then a necessary and sufficient condition for the uniform convergence of the series ∑∞ n=1 a n sin nx...Chaundy and Jolliffe proved that if {a n } is a non-increasing (monotonic) real sequence with lim n →∞ a n = 0, then a necessary and sufficient condition for the uniform convergence of the series ∑∞ n=1 a n sin nx is lim n →∞ na n = 0. We generalize (or weaken) the monotonic condition on the coefficient sequence {a n } in this classical result to the so-called mean value bounded variation condition and prove that the generalized condition cannot be weakened further. We also establish an analogue to the generalized Chaundy-Jolliffe theorem in the complex space.展开更多
In this paper,we develop and analyze a finite difference method for linear second-order stochastic boundary-value problems(SBVPs)driven by additive white noises.First we regularize the noise by the Wong-Zakai approxim...In this paper,we develop and analyze a finite difference method for linear second-order stochastic boundary-value problems(SBVPs)driven by additive white noises.First we regularize the noise by the Wong-Zakai approximation and introduce a sequence of linear second-order SBVPs.We prove that the solution of the SBVP with regularized noise converges to the solution of the original SBVP with convergence order O(h)in the meansquare sense.To obtain a numerical solution,we apply the finite difference method to the stochastic BVP whose noise is piecewise constant approximation of the original noise.The approximate SBVP with regularized noise is shown to have better regularity than the original problem,which facilitates the convergence proof for the proposed scheme.Convergence analysis is presented based on the standard finite difference method for deterministic problems.More specifically,we prove that the finite difference solution converges at O(h)in the mean-square sense,when the second-order accurate three-point formulas to approximate the first and second derivatives are used.Finally,we present several numerical examples to validate the efficiency and accuracy of the proposed scheme.展开更多
In this correspondence,we establish mean convergence theorems for the maximum of normed double sums of Banach space valued random elements.Most of the results pertain to random elements which are M-dependent.We expand...In this correspondence,we establish mean convergence theorems for the maximum of normed double sums of Banach space valued random elements.Most of the results pertain to random elements which are M-dependent.We expand and improve a number of particular cases in the literature on mean convergence of random elements in Banach spaces.One of the main contributions of the paper is to simplify and improve a recent result of Li,Presnell,and Rosalsky[Journal of Mathematical Inequalities,16,117–126(2022)].A new maximal inequality for double sums of M-dependent random elements is proved which may be of independent interest.The sharpness of the results is illustrated by four examples.展开更多
In this paper we discuss the least-square estimator of the unknown change point in a mean shift for moving-average processes of ALNQD sequence. The consistency and the rate of convergence for the estimated change poin...In this paper we discuss the least-square estimator of the unknown change point in a mean shift for moving-average processes of ALNQD sequence. The consistency and the rate of convergence for the estimated change point are established. The asymptotic distribution for the change point estimator is obtained. The results are also true for ρ-mixing, φ-mixing, α-mixing sequences under suitable conditions. These results extend those of Bai, who studied the mean shift point of a linear process of i.i.d, variables, and the condition ∑j=0^∞j｜aj｜ 〈 ∞ in Bai is weakened to ∑j=0^∞｜aj｜〈∞.展开更多
The study explores the asymptotic consistency of the James-Stein shrinkage estimator obtained by shrinking a maximum likelihood estimator. We use Hansen’s approach to show that the James-Stein shrinkage estimator con...The study explores the asymptotic consistency of the James-Stein shrinkage estimator obtained by shrinking a maximum likelihood estimator. We use Hansen’s approach to show that the James-Stein shrinkage estimator converges asymptotically to some multivariate normal distribution with shrinkage effect values. We establish that the rate of convergence is of order and rate , hence the James-Stein shrinkage estimator is -consistent. Then visualise its consistency by studying the asymptotic behaviour using simulating plots in R for the mean squared error of the maximum likelihood estimator and the shrinkage estimator. The latter graphically shows lower mean squared error as compared to that of the maximum likelihood estimator.展开更多
The author obtains the rate of strong convergence,mean squared error and optimal choice of the“smoothing parameter”(the sample fraction)of a tail index estimator which was proposed by the author from Pickands’estim...The author obtains the rate of strong convergence,mean squared error and optimal choice of the“smoothing parameter”(the sample fraction)of a tail index estimator which was proposed by the author from Pickands’estimator,and called modified Pickands’estimator.The similar results about Hill’s estimator are also obtained,which generalize the corresponding results in.Besides,some comparisons between Hill’s estimator and the modified Pickands’estimator are given.展开更多
In this paper,we consider numerical and trigonometric series with a very general monotonicity condition.First,a fundamental decomposition is established from which the sufficient parts of many classical results in Fou...In this paper,we consider numerical and trigonometric series with a very general monotonicity condition.First,a fundamental decomposition is established from which the sufficient parts of many classical results in Fourier analysis can be derived in this general setting.In the second part of the paper a necessary and sufficient condition for the uniform convergence of sine series is proved generalizing a classical theorem of Chaundy and Jolliffe.展开更多
This paper presents a strong predictor-corrector method for the numerical solution of stochastic delay differential equations （SDDEs） of ItS-type. The method is proved to be mean-square convergent of order min{1/2,p...This paper presents a strong predictor-corrector method for the numerical solution of stochastic delay differential equations （SDDEs） of ItS-type. The method is proved to be mean-square convergent of order min{1/2,p} under the Lipschitz condition and the linear growth condition, where p is the exponent of HSlder condition of the initial function. Stability criteria for this type of method are derived. It is shown that for certain choices of the flexible parameter p the derived method can have a better stability property than more commonly used numerical methods. That is, for some p, the asymptotic MS-stability bound of the method will be much larger than that of the Euler-Maruyama method. Numerical results are reported confirming convergence properties and comparing stability properties of methods with different parameters p. Finally, the vectorised simulation is discussed and it is shown that this implementation is much more efficient.展开更多
Recently,Pipoli and Sinestrari[Pipoli,G.and Sinestrari,C.,Mean curvature flow of pinched submanifolds of CPn,Comm.Anal.Geom.,25,2017,799-846]initiated the study of convergence problem for the mean curvature flow of sm...Recently,Pipoli and Sinestrari[Pipoli,G.and Sinestrari,C.,Mean curvature flow of pinched submanifolds of CPn,Comm.Anal.Geom.,25,2017,799-846]initiated the study of convergence problem for the mean curvature flow of small codimension in the complex projective space CPm.The purpose of this paper is to develop the work due to Pipoli and Sinestrari,and verify a new convergence theorem for the mean curvature flow of arbitrary codimension in the complex projective space.Namely,the authors prove that if the initial submanifold in CPm satisfies a suitable pinching condition,then the mean curvature flow converges to a round point in finite time,or converges to a totally geodesic submanifold as t→∞.Consequently,they obtain a differentiable sphere theorem for submanifolds in the complex projective space.展开更多
基金supported by National Sciences and Engineering Research Council of CanadaNational Natural Science Foundation of China (Grant No. 10471130)
文摘Chaundy and Jolliffe proved that if {a n } is a non-increasing (monotonic) real sequence with lim n →∞ a n = 0, then a necessary and sufficient condition for the uniform convergence of the series ∑∞ n=1 a n sin nx is lim n →∞ na n = 0. We generalize (or weaken) the monotonic condition on the coefficient sequence {a n } in this classical result to the so-called mean value bounded variation condition and prove that the generalized condition cannot be weakened further. We also establish an analogue to the generalized Chaundy-Jolliffe theorem in the complex space.
基金partially supported by the NASA Nebraska Space Grant Program and UCRCA at the University of Nebraska at Omaha.
文摘In this paper,we develop and analyze a finite difference method for linear second-order stochastic boundary-value problems(SBVPs)driven by additive white noises.First we regularize the noise by the Wong-Zakai approximation and introduce a sequence of linear second-order SBVPs.We prove that the solution of the SBVP with regularized noise converges to the solution of the original SBVP with convergence order O(h)in the meansquare sense.To obtain a numerical solution,we apply the finite difference method to the stochastic BVP whose noise is piecewise constant approximation of the original noise.The approximate SBVP with regularized noise is shown to have better regularity than the original problem,which facilitates the convergence proof for the proposed scheme.Convergence analysis is presented based on the standard finite difference method for deterministic problems.More specifically,we prove that the finite difference solution converges at O(h)in the mean-square sense,when the second-order accurate three-point formulas to approximate the first and second derivatives are used.Finally,we present several numerical examples to validate the efficiency and accuracy of the proposed scheme.
文摘In this correspondence,we establish mean convergence theorems for the maximum of normed double sums of Banach space valued random elements.Most of the results pertain to random elements which are M-dependent.We expand and improve a number of particular cases in the literature on mean convergence of random elements in Banach spaces.One of the main contributions of the paper is to simplify and improve a recent result of Li,Presnell,and Rosalsky[Journal of Mathematical Inequalities,16,117–126(2022)].A new maximal inequality for double sums of M-dependent random elements is proved which may be of independent interest.The sharpness of the results is illustrated by four examples.
文摘In this paper we discuss the least-square estimator of the unknown change point in a mean shift for moving-average processes of ALNQD sequence. The consistency and the rate of convergence for the estimated change point are established. The asymptotic distribution for the change point estimator is obtained. The results are also true for ρ-mixing, φ-mixing, α-mixing sequences under suitable conditions. These results extend those of Bai, who studied the mean shift point of a linear process of i.i.d, variables, and the condition ∑j=0^∞j｜aj｜ 〈 ∞ in Bai is weakened to ∑j=0^∞｜aj｜〈∞.
文摘The study explores the asymptotic consistency of the James-Stein shrinkage estimator obtained by shrinking a maximum likelihood estimator. We use Hansen’s approach to show that the James-Stein shrinkage estimator converges asymptotically to some multivariate normal distribution with shrinkage effect values. We establish that the rate of convergence is of order and rate , hence the James-Stein shrinkage estimator is -consistent. Then visualise its consistency by studying the asymptotic behaviour using simulating plots in R for the mean squared error of the maximum likelihood estimator and the shrinkage estimator. The latter graphically shows lower mean squared error as compared to that of the maximum likelihood estimator.
文摘The author obtains the rate of strong convergence,mean squared error and optimal choice of the“smoothing parameter”(the sample fraction)of a tail index estimator which was proposed by the author from Pickands’estimator,and called modified Pickands’estimator.The similar results about Hill’s estimator are also obtained,which generalize the corresponding results in.Besides,some comparisons between Hill’s estimator and the modified Pickands’estimator are given.
基金Supported by the European Research Council Advanced Grant(Grant No.267055)
文摘In this paper,we consider numerical and trigonometric series with a very general monotonicity condition.First,a fundamental decomposition is established from which the sufficient parts of many classical results in Fourier analysis can be derived in this general setting.In the second part of the paper a necessary and sufficient condition for the uniform convergence of sine series is proved generalizing a classical theorem of Chaundy and Jolliffe.
基金This work is supported by National Natural Science Foundation of China （Nos. 11401594, 11171125, 91130003） and the New Teachers＇ Specialized Research Fund for the Doctoral Program from Ministry of Education of China （No. 20120162120096）.
文摘This paper presents a strong predictor-corrector method for the numerical solution of stochastic delay differential equations （SDDEs） of ItS-type. The method is proved to be mean-square convergent of order min{1/2,p} under the Lipschitz condition and the linear growth condition, where p is the exponent of HSlder condition of the initial function. Stability criteria for this type of method are derived. It is shown that for certain choices of the flexible parameter p the derived method can have a better stability property than more commonly used numerical methods. That is, for some p, the asymptotic MS-stability bound of the method will be much larger than that of the Euler-Maruyama method. Numerical results are reported confirming convergence properties and comparing stability properties of methods with different parameters p. Finally, the vectorised simulation is discussed and it is shown that this implementation is much more efficient.
基金supported by the National Natural Science Foundation of China(Nos.12071424,11531012,12201087).
文摘Recently,Pipoli and Sinestrari[Pipoli,G.and Sinestrari,C.,Mean curvature flow of pinched submanifolds of CPn,Comm.Anal.Geom.,25,2017,799-846]initiated the study of convergence problem for the mean curvature flow of small codimension in the complex projective space CPm.The purpose of this paper is to develop the work due to Pipoli and Sinestrari,and verify a new convergence theorem for the mean curvature flow of arbitrary codimension in the complex projective space.Namely,the authors prove that if the initial submanifold in CPm satisfies a suitable pinching condition,then the mean curvature flow converges to a round point in finite time,or converges to a totally geodesic submanifold as t→∞.Consequently,they obtain a differentiable sphere theorem for submanifolds in the complex projective space.
