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central limit theorem AND CONVERGENCE RATES FOR A SUPERCRITICAL BRANCHING PROCESS WITH IMMIGRATION IN A RANDOM ENVIRONMENT

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Acta Mathematica Scientia 2022年第3期42卷 957-974页

作者： Yingqiu LI Xulan HUANG Zhaohui PENG Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering School of Mathematics and StatisticsChangsha University of Science and TechnologyChangsha 410004China School of Mathematics and Statistics Changsha University of Science and TechnologyChangsha 410004China

We are interested in the convergence rates of the submartingale Wn=Z_(n)/Π_(n)to its limit W,where(Π_(n))is the usually used norming sequence and(Z_(n))is a supercritical branching process with immigration(Y_(n))in a stationary and ergodic environmentξ.Under suitable conditions,we establish the following central limit theorems and results about the rates of convergence in probability or in law:(i)W-W_(n) with suitable normalization converges to the normal law N(0,1),and similar results also hold for W_(n+k)-W_(n) for each fixed k∈N^(*);(ii)for a branching process with immigration in a finite state random environment,if W_(1) has a finite exponential moment,then so does W,and the decay rate of P(|W-W_(n)|>ε)is supergeometric;(iii)there are normalizing constants an(ξ)(that we calculate explicitly)such that a_(n)(ξ)(W-W_(n))converges in law to a mixture of the Gaussian law.

关键词： Branching process with immigration random environment convergence rates central limit theorem convergence in law convergence in probability

Multidimensional Local central limit theorem of Some Non-uniformly Hyperbolic Systems

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Acta Mathematica Sinica,English Series 2009年第4期25卷 565-580页

作者： Hong Qiang XIA College of Science Wuhan University of Science and Engineering No. 1 Fangzhi Road Wuhan 430073 P. R. China

We consider Young＇s nonuniformly hyperbolic system （X, T, u） where u is the SRB measure corresponding to the system （X, T）, and show that if the components of a Holder observable f ： X → R^d are cohomologously independent, then f satisfies the multidimensional central limit theorem. Moreover if f is aperiodic, then f satisfies the local multidimensional central limit theorem.

关键词： central limit theorem transfer operator nonuniform hyperbolic diIQ[eomorphism

A central limit theorem for m-dependent Random Variables under Sublinear Expectations

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Acta Mathematicae Applicatae Sinica 2015年第2期31卷 435-444页

作者： Xin-peng LI School of Mathematics Shandong University

In this paper, we prove a central limit theorem for m-dependent random variables under sublinear expectations. This theorem can be regarded as a generalization of Peng＇s central limit theorem.

关键词： central limit theorem m-dependence G-normal distribution

Asymptotic expansions in the central limit theorem for a branching Wiener process

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Science China Mathematics 2021年第12期64卷 2759-2774页

作者： Zhi-Qiang Gao Quansheng Liu Laboratory of Mathematics and Complex Systems(Ministry of Education) School of Mathematical SciencesBeijing Normal UniversityBeijing 100875China LMBA Campus de TohannicUniversite de Bretagne-SudCNRS UMR 6205Vannes F-56000France

We consider a branching Wiener process in R^(d),in which particles reproduce as a super-critical Galton-Watson process and disperse according to a Wiener *** B⊂R^(d),let Z_(n)(B) be the number of particles of generation n located in *** study of the central limit theorem and related results about the counting measure Z_(n)(·)is important because such results give good descriptions of the con guration of the branching Wiener process at time *** earlier works,the exact convergence rate in the central limit theorem and the asymptotic expansion until the third order have been ***,we establish the asymptotic expansion of any order in the central limit theorem under a moment condition of the form EX(logX)^(1+λ)<∞.

关键词： branching Wiener process asymptotic expansion central limit theorem

THE SPHERICAL FUNCTIONS AND THE central limit theorem ON SU(2,2)/S(U(2)×U(2))

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Acta Mathematica Scientia 2007年第4期27卷 867-874页

作者：杨乔华张钦 Wuhan Institute of Physics and Mathematics Chinese Academy of Sciences Wuhan 430071China School of Mathematics and Statistics Wuhan University Wuhan 430072China

Let G = SU（2, 2）, K = S（U（2） × U（2））, and for l ∈ Z, let {Tl}l∈z be a one-dimensional K-type and let El be the line bundle over G/K associated to Tl. It is shown that the Tl-spherical function on G is given by the hypergeometric functions of several variables. By applying this result, a central limit theorem for the space G/K is obtained.

关键词： Spherical function central limit theorem hypergeometric function

Functional central limit theorem for super α-stable processes

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Science China Mathematics 2004年第6期47卷 874-881页

作者： HONG WenmingDepartment of Mathematics, Beijing Normal University, Beijing 100875, China Department of Mathematics Beijing Normal University Beijing China

A functional central limit theorem is proved for the centered occupation time process of the super α-stable processes in the finite dimensional distribution sense. For the intermediate dimensions α limiting process is a Gaussian process, whose covariance is specified; for the critical dimension d= 2α and higher dimensions d limiting process is Brownian motion.

关键词： super α-stable processes occupation time central limit theorem evolution equation

A central limit theorem FOR STRONG NEAR-EPOCH DEPENDENT RANDOM VARIABLES

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Chinese Annals of Mathematics,Series B 2004年第2期25卷 263-274页

作者： LINZHENGYAN QIUJIN DepartmentofMathematics Zhe.iiangUniversity（XixiCampus）Hangzhou310028China. DepartmentofMathematics ZhejiangUniversity（XixiCampus）Hangzhou310028China.

In this paper, a central limit theorem for strong near-epoch dependent sequences of random variables introduced in [9] is showed. Under the same moments condition,the authors essentially weaken the "size" requirement ... 详细信息

关键词： Mixingale Mixing Near-epoch dependent Strong near-epoch dependent central limit theorem

On necessary and sufficient conditions for the self-normalized central limit theorem In Honor of Professor Chuanrong Lu on His 85th Birthday

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Science China Mathematics 2018年第10期61卷 1741-1748页

作者： Qiman Shao Department of Statistics The Chinese University of Hong Kong

Let X_1, X_2,... be a sequence of independent random variables and S_n=sum X_1 from i=1 to n and V_n^2=sum X_1~2 from i=1 to n . When the elements of the sequence are i.i.d., it is known that the self-normalized sum S_n/V_n converges to a standard normal distribution if and only if max1≤i≤n|X_i|/V_n → 0 in probability and the mean of X_1 is zero. In this paper, sufficient conditions for the self-normalized central limit theorem are obtained for general independent random variables. It is also shown that if max1≤i≤n|X_i|/V_n → 0 in probability, then these sufficient conditions are necessary.

关键词： central limit theorem self-normalized independent random variables

Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth

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Frontiers of Mathematics in China 2018年第4期13卷 913-933页

作者： Yongqiang SUO Jin TAO Wei ZHANG School of Mathematics and Statistics Central South University Changsha 410083 China Department of Mathematics Swansea University Singleton Park SA2 8PP UK Department of Mathematics Southern University of Science and Technology Shenzhen 518055 China

Employing the weak convergence method, based on a variational representation for expected values of positive functionals of a Brownian motion, we investigate moderate deviation for a class of stochastic differential delay equations with small noises, where the coefficients are allowed to be highly nonlinear growth with respect to the variables. Moreover, we obtain the central limit theorem for stochastic differential delay equations which the coefficients are polynomial growth with respect to the delay variables.

关键词： Stochastic differential delay equation (SDDE) polynomial growth central limit theorem moderate deviation principle weak convergence