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THE LONG TIME BEHAVIOR OF THE FRACTIONAL ORNSTEIN-UHLENBECK PROCESS WITH LINEAR SELF-REPELLING DRIFT

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Acta Mathematica Scientia 2024年第2期44卷 671-685页

作者：夏晓宇闫理坦杨晴 College of Information Science and Technology Donghua UniversityShanghai201620China Department of Statistics College of ScienceDonghua UniversityShanghai201620China

Let B^(H) be a fractional Brownian motion with Hurst index 1/2≤H<1.In this paper,we consider the equation(called the Ornstein-Uhlenbeck process with a linear self-repelling drift)dX_(t)^(H)=dB_(t)^(H)+σ X_(t)^(H)dt+vdt-θ(∫_(0)^(t)(X_(t)^(H)-X_(s)^(H))ds)dt,whereθ<0,σ,v∈ℝ.The process is an analogue of self-attracting diffusion(Cranston,Le Jan.Math Ann,1995,303:87–93).Our main aim is to study the large time behaviors of the process.We show that the solution X^(H)diverges to infinity as t tends to infinity,and obtain the speed at which the process X^(H)diverges to infinity.

关键词： fractional Brownian motion stochastic difference equations rate of convergence asymptotic

On the rate of convergence of two generalized Bernstein type operators

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Applied Mathematics(A Journal of Chinese Universities) 2020年第3期35卷 321-331页

作者： LIAN Bo-yong CAI Qing-bo Department of Mathematics Yang-En UniversityQuanzhou 362014China School of Mathematics and Computer Science Quanzhou Normal UniversityQuanzhou 362000China

In this paper,we introduce the Bézier variant of two new families of generalized Bernstein type operators.We establish a direct approximation by means of the Ditzian-Totik modulus of smoothness and a global approximation theorem in terms of second order modulus of continuity.By means of construction of suitable functions and the method of Bojanic and Cheng,we give the rate of convergence for absolutely continuous functions having a derivative equivalent to a bounded variation function.

关键词： Bernstein operators modulus of smoothness rate of convergence bounded variation

Estimation of convergence rate for multi-regression learning algorithm

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Science China(Information Sciences) 2012年第3期55卷 701-713页

作者： XU ZongBen1,2, ZHANG YongQuan3,1,2 & CAO FeiLong3 1Institute for Information and System Sciences, Xi’an Jiaotong University, Xi’an 710049, China 2MOE Key Labratory for Intelligent Networks and Network Security, Xi’an Jiaotong University, Xi’an 710049, China 3Department of Information and Mathematics Sciences, China Jiliang University, Hangzhou 310018, China Institute for Information and System Sciences Xi’an Jiaotong University Xi’an China MOE Key Labratory for Intelligent Networks and Network Security Xi’an Jiaotong University Xi’an China Department of Information and Mathematics Sciences China Jiliang University Hangzhou China

In many applications, the pre-information on regression function is always unknown. Therefore, it is necessary to learn regression function by means of some valid tools. In this paper we investigate the regression problem in learning theory, i.e., convergence rate of regression learning algorithm with least square schemes in multi-dimensional polynomial space. Our main aim is to analyze the generalization error for multi-regression problems in learning theory. By using the famous Jackson operators in approximation theory, covering number, entropy number and relative probability inequalities, we obtain the estimates of upper and lower bounds for the convergence rate of learning algorithm. In particular, it is shown that for multi-variable smooth regression functions, the estimates are able to achieve almost optimal rate of convergence except for a logarithmic factor. Our results are significant for the research of convergence, stability and complexity of regression learning algorithm.

关键词： learning theory covering number rate of convergence entropy number

convergence rate of Peng’s law of large numbers under sublinear expectations

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Probability, Uncertainty and Quantitative Risk 2021年第3期6卷 261-266页

作者： Mingshang Hu Xiaojuan Li Xinpeng Li Zhongtai Securities Institute for Financial Studies Shandong UniversityJinan 250100China Research Center for Mathematics and Interdisciplinary Sciences Shandong UniversityQingdao 266237China

This short note provides a new and simple proof of the convergence rate for the Peng’s law of large numbers under sublinear expectations,which improves the results presented by Song[15]and Fang et al.[3].

关键词： Law of large numbers rate of convergence Sublinear expectation

rate of convergence of Normal Approximation for Local Median Estimate

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Tsinghua Science and Technology 1998年第4期3卷 1213-1217页

作者：杨瑛 Department of Applied Mathematics Tsinghua University Beijing 100084

Consider the fixed\|design nonparametric median regression model Y \{ni =g(x \{ni )+ε \{ni ,1≤i≤n, where ε \{ni are iid random variables with median zero. In estimating the regression function g(x), local median estimates \{nh (x) are employed, where h is the number of neighbors of x. Under some regularity conditions, the asymptotic normality and rate of convergence of normal approximation are obtained.

关键词： asymptotic normality local median estimate nonparametric median regression rate of convergence

ON THE rate of convergence OF SOLUTIONS FOR A CLASS OF DIFFERENCE EQUATIONS

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Annals of Differential Equations 2000年第3期16卷 221-227页

作者：陈武华

Consider the following difference equation x(n) - x(n - 1) = p(n)G(x(n), x(n - k(n))) (1)where p(n)≤ M, G(x, .) is non-decreasing, G(., y) is non-increasing and G(x, x) = 0. Under some condition, we prove that every solution of (1) tends to constant and an estimate of the rate of solutions of (1) is given, the main results of paper [4] are extended.

关键词： difference equation rate of convergence

A Survey on Some Recent Developments of Alternating Direction Method of Multipliers

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Journal of the Operations Research Society of China 2022年第1期10卷 1-52页

作者： De-Ren Han LMIB of the Ministry of Education School of Mathematical SciencesBeihang UniversityBeijing 100191China

Recently, alternating direction method of multipliers (ADMM) attracts much attentions from various fields and there are many variant versions tailored for differentmodels. Moreover, its theoretical studies such as rate of convergence and extensionsto nonconvex problems also achieve much progress. In this paper, we give a surveyon some recent developments of ADMM and its variants.

关键词： Alternating direction method of multipliers Global convergence rate of convergence Nonconvex optimization

Approximation by Sz´asz Type Operators Involving Apostol-Genocchi Polynomials

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Computer Modeling in Engineering & Sciences 2022年第1期130卷 287-297页

作者： Mine Menekse Yılmaz Department of Mathematics Faculty of Arts and ScienceUniversity of GaziantepGaziantepTR-27310Turkey

The goal of this paper is to give a form of the operator involving the generating function of Apostol-Genocchi polynomials of orderα.Applying the Korovkin theorem,we arrive at the convergence of the operator with the aid of moments and central moments.We determine the rate of convergence of the operator using several tools such as K-functional,modulus of continuity,second modulus of continuity.We also give a type of Voronovskaya theorem for estimating error.Moreover,we investigate some results about convergence properties of the operator in a weighted space.Finally,we give numerical examples to support our theorems by using the Maple.

关键词： Apostol-Genocchi polynomials rate of convergence Korovkin theorem modulus of continuity Sz´asz type operators generating functions

Propagation of chaos and conditional McKean-Vlasov SDEs with regime-switching

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Frontiers of Mathematics in China 2022年第4期17卷 731-746页

作者： Jinghai SHAO Dong WEI Center for Applied Mathematics Tianjin UniversityTianjin300072China

We investigate a particle system with mean field interaction living in a random environment characterized by a regime-switching process.The switching process is allowed to be dependent on the particle system.The well-posedness and various properties of the limit conditional McKean-Vlasov SDEs are studied,and the conditional propagation of chaos is established with explicit estimate of the convergence rate.

关键词： Regime-switching propagation of chaos Wasserstein distance conditional McKean-Vlasov SDEs rate of convergence