CENTRAL LIMIT THEOREM AND CONVERGENCE RATES FOR A SUPERCRITICAL BRANCHING PROCESS WITH IMMIGRATION IN A RANDOM ENVIRONMENT

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.

RATE OF CONVERGENCE AND EXPANSION OF RNYI ENTROPIC CENTRAL LIMIT THEOREM

We obtain the expansion of Renyi divergence of order α （0 〈 α 〈 1） between the normalized sum of IID continuous random variables and the Caussian limit under minimal moment conditions via Edgeworth-type expansion. The rate is faster than that of Shannon case, which can be used to improve the rate of convergence in total variance norm.

THE DIVERGENCE-CONVERGENCE EFFECT OF SURFACE CURRENT AND ITS ACTION ON THE BOHAI SEA

The action of the wind field and the influence of topography can cause divergence or convergence of surface current. The existence of the divergence-convergence effect is proved and the dynamical significance of the divergent or convergent state and its link with many marine phenomena are pointed out. Divergence fields of surface current in the Bohai Sea in winter and summer are obtained by numerical modelling describing the divergence-convergence character of seasonally wind-driven current. The relation between the effect and seasonal marine phenomena is discussed. Study on the divergence-convergence effect of surface current (DCESC)can be an indirect method for testing the calculated results.

ON THE CHOICES OF ACCELERATING CONVERGENCE FACTORS FOR LIMIT PERIODIC CONTINUED FRACTION K(an/1)

There are many accelerating convergence factors (ACFs) for limit periodic continued fraction K(an/1)(an→a≠0). In this paper, some characteristics and comparative theorems are ob tained on ACFs. Two results are given for most frequently used ACFs.

The Convergence Rate of Ergodic Limits for Regularized Resolvent Families

This paper is concerned with the convergence rates of ergodic limits and approximation for regularized resolvent families for a linear Volterra integral equation. The results contain C 0-semigroups, cosine operator functions and α-times integrated resolvent family as special cases.

A NOTE ON EXACT CONVERGENCE RATE IN THE LOCAL LIMIT THEOREM FOR A LATTICE BRANCHING RANDOM WALK

Consider a branching random walk, where the underlying branching mechanism is governed by a Galton-Watson process and the moving law of particles by a discrete random variable on the integer lattice Z. Denote by Zn（z） the number of particles in the n-th generation in the model for each z ∈ Z. We derive the exact convergence rate in the local limit theorem for Zn（z） assuming a condition like ＂EN（logN）1＋λ 〈 ∞＂ for the offspring distribution and a finite moment condition on the motion law. This complements the known results for the strongly non-lattice branching random walk on the real line and for the simple symmetric branching random walk on the integer lattice.

Technology Convergence and Divergence in China’s Electronic Industries:Evidence from Patents

By thoroughly reviewing international studies on technology convergence and divergence, four kinds of hypothesis are proposed based on patent data Herfindhal index （HI） measurement. The main fmding is that technology convergence does exist, based on patent technology records in China, primarily driven by overseas companies＇ strategic behavior, such as field intensiveness, competition during technology maturity session, and patent technology growth.

Investigation on the effect of beam divergence angle upon output waveform based on stimulated Brillouin scattering optical limiting

This paper investigates the effect of beam divergence angle on output waveform based on stimulated Brillouin scattering optical limiting.Output waveforms in the case of different pump divergence angles are numerically simulated,and validated in a Nd：YAG seed-injected laser system.The results indicate that a small pump divergence angle can lead to good interaction between pump and Stokes,and a platform can be easily realized in the transmitted waveform.In contrast,a peak followed by the platform appears when the divergence angle becomes large.

Iterative and adjusting method for computing stream function and velocity potential in limited domains and convergence analysis

The stream function and the velocity potential can be easily computed by solving the Poisson equations in a unique way for the global domain. Because of the var- ious assumptions for handling the boundary conditions, the solution is not unique when a limited domain is concerned. Therefore, it is very important to reduce or eliminate the effects caused by the uncertain boundary condition. In this paper, an iterative and ad- justing method based on the Endlich iteration method is presented to compute the stream function and the velocity potential in limited domains. This method does not need an explicitly specifying boundary condition when used to obtain the effective solution, and it is proved to be successful in decomposing and reconstructing the horizontal wind field with very small errors. The convergence of the method depends on the relative value for the distances of grids in two different directions and the value of the adjusting factor. It is shown that applying the method in Arakawa grids and irregular domains can obtain the accurate vorticity and divergence and accurately decompose and reconstruct the original wind field. Hence, the iterative and adjusting method is accurate and reliable.

ON THE RATES OF CONVERGENCE IN THE CENTRAL LIMIT THEOREM FOR TWO－PARAMETER MARTINGALE DIFFERENCES

In this paper we obtain the uniform bounds on the rate of convergence in the central limit theorem (CLT) for a class of two-parameter martingale difference sequences under certain conditions.

Impact of the central Pacific zonal wind divergence and convergence on the central Pacific El Nio event

The central Pacific（CP） zonal wind divergence and convergence indices are defined, and the forming mechanism of CP El Nio（La Nia） events is discussed preliminarily. The results show that the divergence and convergence of the zonal wind anomaly（ZWA） are the key process in the forming of CP El Nio（La Nia） events. A correlation analysis between the central Pacific zonal wind divergence and convergence indices and central Pacific El Nio indices indicates that there is a remarkable lag correlation between them. The central Pacific zonal wind divergence and convergence indices can be used to predict the CP events. Based on these results, a linear regression equation is obtained to predict the CP El Nio（La Nia） events 5 months ahead.

Exact Convergence Rate of the Local Limit Theorem for a Branching Random Walk in Z^(d)with a Random Environment in Time

Consider a branching random walk with a random environment in time in the d-dimensional integer lattice.The branching mechanism is governed by a supercritical branching process,and the particles perform a lazy random walk with an independent,non-identical increment distribution.For A■Z^(d),let Z_(n)(A)be the number of offsprings of generation n located in A.The exact convergence rate of the local limit theorem for the counting measure Z_(n)(·)is obtained.This partially extends the previous results for a simple branching random walk derived by Gao(2017,Stoch.Process Appl.).

Characterizing Continuous dcpos by Liminf Convergence of Filters

It is proved in this note that,under a mild assumption,a dcpo L is continuous if and only if the liminf convergence on L is topological.

GENERALIZED MATRIX MULTISPLITTING RELAXATION METHODS AND THEIR CONVERGENCE

In this paper, we set up a general framework of parallel matrix mullisplitting relaxation methods for solving large scale system of linear equations. We investigate the convergence properties of this framework and give several sufficient conditions ensuring it to converge as well as diverge. At last, we conclude a necessary and sufficient condition for the convergence of this framework when the coefficient matrix is an L-matrix.

THE INVISCID AND NON-RESISTIVE LIMIT IN THE CAUCHY PROBLEM FOR 3-D NONHOMOGENEOUS INCOMPRESSIBLE MAGNETO-HYDRODYNAMICS

In this paper,the inviscid and non-resistive limit is justified for the local-in-time solutions to the equations of nonhomogeneous incompressible magneto-hydrodynamics （MHD）in R3.We prove that as the viscosity and resistivity go to zero,the solution of the Cauchy problem for the nonhomogeneous incompressible MHD system converges to the solution of the ideal MHD system.The convergence rate is also obtained simultaneously.

An Adaptive Load Stepping Algorithm for Path-Dependent Problems Based on Estimated Convergence Rates

A new adaptive(automatic)time stepping algorithm,called RCA(Rate of Convergence Algorithm)is presented.The new algorithm was applied in nonlinear finite element analysis of path-dependent problems.The step size is adjusted by monitoring the estimated convergence rate of the nonlinear iterative process.The RCA algorithm is relatively simple to implement,robust and its performance is comparable to,and in some cases better than,the automatic load incrementaion algorithm existent in commercial codes.Discussions about the convergence rate of nonlinear iterative processes,an estimation of the rate and a study of the parameters of the RCA algorithm are presented.To show the capacity of the algorithm to adjust the increment size,detailed discussions based on results for different limit load analyses are presented.The results obtained by RCA algorithm are compared with those by ABAQUS?,one of the most powerful nonlinear FEA(Finite Element Analysis)commercial software,in order to verify the capability of RCA algorithm to adjust the increment size along nonlinear analyses.

BOUNDARY LAYER AND VANISHING DIFFUSION LIMIT FOR NONLINEAR EVOLUTION EQUATIONS

In this paper, we consider an initial-boundary value problem for some nonlinear evolution equations with damping and diffusion. The main purpose is to investigate the boundary layer effect and the convergence rates as the diffusion parameter α goes to zero.

Global Convergence of a Modified Limited Memory BFGS Method for Non-convex Minimization

In this paper, a modified limited memory BFGS method for solving large-scale unconstrained optimization problems is proposed. A remarkable feature of the proposed method is that it possesses global convergence property without convexity assumption on the objective function. Under some suitable conditions, the global convergence of the proposed method is proved. Some numerical results are reported which illustrate that the proposed method is efficient.

A THEOREM ON THE CONVERGENCE OF SUMS OF INDEPENDENT RANDOM VARIABLES

Let Sigma (infinity)(n=1) X-n be a series of independent random variables with at least one non-degenerate X-n, and let F-n be the distribution function of its partial sums S-n = Sigma (n)(k=1) X-k. Motivated by Hildebrand's work in [1], the authors investigate the a.s. convergence of Sigma (infinity)(n=1) X-n under a hypothesis that Sigma (infinity)(n=1) rho (X-n, c(n)) = infinity whener Sigma (infinity)(n=1) c(n) diverges, where the notation rho (X,c) denotes the Levy distance between the random variable X and the constant c. The principal result of this paper shows that the hypothesis is the condition under which the convergence of F-n(x(0)) with the limit value 0 < L-0 < 1, together with the essential convergence of Sigma (infinity)(n=1) X-n, is both sufficient and necessary in order for the series Sigma (infinity)(n=1) X-n to a.s. coverage. Moreover, if the essential convergence of Sigma (infinity)(n=1) X-n is strengthened to limsup(n=infinity) P(\S-n\ < K) = 1 for some K > 0, the hypothesis is already equivalent to the a.s. convergence of Sigma (infinity)(n=1) X-n. Here they have not only founded a very general limit theorem, but improved the related result in Hildebrand([1]) as well.

A Modified Limited SQP Method For Constrained Optimization

In this paper, a modified variation of the Limited SQP method is presented for constrained optimization. This method possesses not only the information of gradient but also the information of function value. Moreover, the proposed method requires no more function or derivative evaluations and hardly more storage or arithmetic operations. Under suitable conditions, the global convergence is established.