CENTRAL LIMIT THEOREMS FOR A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME

We consider a branching random walk with a random environment m time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environment indexed by the time n. The envi- ronment is supposed to be independent and identically distributed. For A C R, let Zn（A） be the number of particles of generation n located in A. We show central limit theorems for the counting measure Zn （-） with appropriate normalization.

SOME CENTRAL LIMIT THEOREMS FOR SUPER BROWNIAN MOTION

The author proves a central limit theorem for the critical super Brownian motion, which leads to a Gaussian random field. In the transient case the limiting field is the same aa that obtained by Dawson (1977). In the recurrent case it is a spatially uniform field. The author also give a central limit theorem for the weighted occupation time of the super Brownian motion with underlying dimension number d less than or equal to 3, completing the results of Iscoe (1986).

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.

LOCAL CENTRAL LIMIT THEOREM AND BERRY-ESSEEN THEOREM FOR SOME NONUNIFORMLY HYPERBOLIC DIFFEOMORPHISMS

We prove that, for non-uniformly hyperbolic diffeomorphisms in the sense of Young, the local central limit theorem holds, and the speed in the central limit theorem is O（1/√n）.

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.

A MULTIDIMENSIONAL CENTRAL LIMIT THEOREM WITH SPEED OF CONVERGENCE FOR AXIOM A DIFFEOMORPHISMS

Let T：X → X be an Axiom A diffeomorphism,m the Gibbs state for a Hlder continuous function ɡ. Assume that f：X → R^d is a Hlder continuous function with ∫_X^（fdm） = 0.If the components of f are cohomologously independent, then there exists a positive definite symmetric matrix σ~2：=σ~2 （f ） such that S^fn √ n converges in distribution with respect to m to a Gaussian random variable with expectation 0 and covariance matrix σ~2 . Moreover, there exists a real number A 〉 0 such that, for any integer n ≥ 1,Π（ m＊（ 1√ nS f n ）,N （0,σ~2 ） ≤A√n, where m＊（1√ n S^fn）denotes the distribution of 1√ n S^fn with respect to m, and Π is the Prokhorov metric.

The central limit theorem and chaos

Let X be a compact metric space studies some relationships between stochastic and f ： X→ X be a continuous map. This paper and topological properties of dynamical systems. It is shown that if f satisfies the central limit theorem, then f is topologically ergodic and / is sensitively dependent on initial conditions if and only if / is neither minimal nor equicontinuous.

THE SPHERICAL FUNCTIONS AND THE CENTRAL LIMIT THEOREM ON SU(2,2)/S(U(2)×U(2))

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.

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.

A Note on the Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Partial Sums of Certain Stationary Gaussian Sequences

Considering a sequence of standardized stationary Gaussian random variables, a universal result in the almost sure central limit theorem for maxima and partial sum is established. Our result generalizes and improves that on the almost sure central limit theory previously obtained by Marcin Dudzinski [1]. Our result reaches the optimal form.

ON THE CENTRAL LIMIT THEOREM IN PRODUCT SPACES

Suppose that E and F are separable Banach spaces, X and Y are independent symmetric E and F-valued random vectors respectively. This paper is devoted to the study of the central limit theorem for X Y in the injective and projective tensor product spaces E F and E F. Special attention is paid to l2 l2. In addition, two counter-examples are given.

An Almost Sure Central Limit Theorem for Weighted Sums of Mixing Sequences

In this paper, we prove an almost sure central limit theorem for weighted sums of mixing sequences of random variables without stationary assumptions. We no longer restrict to logarithmic averages, but allow rather arbitrary weight sequences. This extends the earlier work on mixing random variables

The Central Limit Theorem on SU(n,1)/S(U(n)×U(1))

Let G = SU（n, 1）, K = S（U（n） × U（1））, and for l ∈Z, let {T;},l∈Z be a one- Dimensional K-type and let Et be the line bundle over G/K associated to Tl. In this work we obtain a central limit theorem for the space Et.

General Operational Protocol for Coherence. Central Limit Theorem as Approximation

A general operational protocol which provides permanent macroscopic coherence of the response of any stable complex system put in an ever-changing environment is proposed. It turns out that the coherent response consists of two parts: 1) a specific discrete pattern, called by the author homeostatic one, whose characteristics are robust to the statistics of the environment;2) the rest part of the response forms a stationary homogeneous process whose coarse-grained structure obeys universal distribution which turns out to be scale-invariant. It is demonstrated that, for relatively short time series, a measurement, viewed as a solitary operation of coarse-graining, superimposed on the universal distribution results in a rich variety of behaviors ranging from periodic-like to stochastic-like, to a sequences of irregular fractal-like objects and sequences of random-like events. The relevance of the Central Limit theorem applies to the latter case. Yet, its application is still an approximation which holds for relatively short time series and for specific low resolution of the measurement equipment. It is proven that the asymptotic behavior in each and every of the above cases is provided by the recently proven decomposition theorem.

A Note on the Almost Sure Central Limit Theorem for Partial Sums of ρ^(−)-Mixing Sequences

Let be a strictly stationary sequence of ρ?-mixing random variables. We proved the almost sure central limit theorem, containing the general weight sequences, for the partial sums , where , . The result generalizes and improves the previous results.

A FUNCTIONAL CENTRAL LIMIT THEOREM FOR NEGATIVELY ASSOCIATED SEQUENCE

Let {X j,j1} be a sequence of negatively associated random variables with EX j=0,EX 2 j【∞. In this paper a functional central limit theorem for negatively associated random variables under some conditions without stationarity is proved, which is the same as the results for positively associated random variables.

基于MATLAB的De Moivre-Laplace中心极限定理的随机模拟

针对De Moivre-Laplace中心极限定理,首先利用特征函数和独立同分布下的中心极限定理得到理论上证明,然后利用MATLAB软件随机模拟分析二项分布与正态分布之间的关系,并对图像进行误差分析,最后证明数值结果与实验结果的一致性。通过这两种不同角度的论证,给出De MoivreLaplace中心极限定理直观的解释和说明,这种将数学实验与数学原理结合的方法,不仅使数学原理化抽象为具体,便于理解和实践,而且扩展了数学原理和计算机软件的应用。

Almost sure central limit theory for products of sums of partial sums

Consider a sequence of i.i.d.positive random variables.An universal result in almost sure limit theorem for products of sums of partial sums is established.We will show that the almost sure limit theorem holds under a fairly general condition on the weight dk= k-1 exp（lnβk）,0≤β〈1.And in a sense,our results have reached the optimal form.

LIMIT THEOREMS FOR A GALTON-WATSON PROCESS IN THE I.I.D. RANDOM ENVIRONMENT

In this article, we obtain the central limit theorem and the law of the iterated logarithm for Galton-Watson processes in i.i.d, random environments.

Almost sure limit theorem for the maximum of a class of quasi-stationary sequences

This paper investigates the problem of almost sure limit theorem for the maximum of quasi-stationary sequence based on the result of Turkman and Walker. We prove an almost sure limit theorem for the maximum of a class of quasi-stationary sequence under weak dependence conditions of D （uk, un） and αtm,ln = 0 （（log log n）-（1＋ε））.