U-统计量的几乎处处中心极限定理

本文得到了U-统计量的几乎处处中心极限定理(ASCLT).在EX1=0,EX2=1下,Berkes等[7]在一定条件下获得了i.i.d.随机变量序列部分和的函数型ASCLT,本文在同样的条件下取得了类似的结果     

Almost Sure Central Limit Theorems for Heavily Trimmed Sums

We obtain an almost sure central limit theorem (ASCLT) for heavily trimmed sums. We also prove a function-typed ASCLT under the same conditions that assure measurable functions to satisfy the ASCLT for the partial sums of i.i.d, random variables with E X1 = 0, EX1^2 = 1.     

A note on the ASCLT for triangular arrays of random variables with an extension to U-statistics

In this paper, we obtain an almost sure central limit theorem for products of independent sums of positive random variables. An extension of the result gives an ASCLT for the U-statistics.     

On the Almost Sure Central Limit Theorem for a Class of Z d -Actions

As an extension of earlier papers on stationary sequences, a concept of weak dependence for strictly stationary random fields is introduced in terms of so-called homoclinic transformations. Under assumptions made within the framework of this concept a form of the almost sure central limit theorem (ASCLT) is established for random fields arising from a class of algebraic Z ^d -actions on compact abelian groups. As an auxillary result, the central limit theorem is proved via Ch. Stein's method. The next stage of the proof includes some estimates which are specific for ASCLT. Both steps are based on making use of homoclinic transformations.     

The Intersective ASCLT

Abstract

In a recent work of the second named author on the Almost Sure Central Limit Theorem (ASCLT), we showed the usefulness of the concept of quasi-orthogonal system of random variables introduced by Bellman and later developed by Kac, Salem and Zygmund. In this paper, we propose an optimal formulation of the ASCLT again by using this idea and new correlation inequalities for sums of independent random variables. We also introduce and develop the notion of “intersective ASCLT” by proving some new results generalizing and improving substancially the classical formulation of the ASCLT. Essential tools for this approach are correlation inequalities recently developed by the first named author and some extensions of these ones obtained in the present paper.     

An almost sure central limit theorem for self-normalized weighted sums

Let X, X1 , X2 , ··· be a sequence of nondegenerate i.i.d. random variables with zero means, which is in the domain of attraction of the normal law. Let {a ni , 1≤i≤n, n≥1} be an array of real numbers with some suitable conditions. In this paper, we show that a central limit theorem for self-normalized weighted sums holds. We also deduce a version of ASCLT for self-normalized weighted sums.     

Escape times for branching processes with random mutational fitness effects

We consider a large declining population of cells under an external selection pressure, modeled as a subcritical branching process. This population has genetic variation introduced at a low rate which leads to the production of exponentially expanding mutant populations, enabling population escape from extinction. Here we consider two possible settings for the effects of the mutation: Case (I) a deterministic mutational fitness advance and Case (II) a random mutational fitness advance. We first establish a functional central limit theorem for the renormalized and sped up version of the mutant cell process. We establish that in Case (I) the limiting process is a trivial constant stochastic process, while in Case (II) the limit process is a continuous Gaussian process for which we identify the covariance kernel. Lastly we apply the functional central limit theorem and some other auxiliary results to establish a central limit theorem (in the large initial population limit) of the first time at which the mutant cell population dominates the population. We find that the limiting distribution is Gaussian in both Cases (I) and (II), but a logarithmic correction is needed in the scaling for Case (II). This problem is motivated by the question of optimal timing for switching therapies to effectively control drug resistance in biomedical applications.     

Central Limit Theorems for Supercritical Superprocesses with Immigration

In this paper, we establish a central limit theorem for a large class of general supercritical superprocesses with immigration with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem extends and generalizes the results obtained by Ren et al. (Stoch Process Appl 125:428–457, 2015). We first give laws of large numbers for supercritical superprocesses with immigration since there are few convergence results on immigration superprocesses, then based on these results, we establish the central limit theorem.     

The central limit theorem for a sequence of random processes with space-varying long memory*

In this paper, we investigate a sequence of square-integrable random processes with space-varying memory. We establish sufficient conditions for the central limit theorem in the space L ²(μ) for the partial sums of the sequence of random processes with space-varying long memory. Of particular interest is a nonstandard normalization of the partial sums in the central limit theorem.     

Central limit theorems for supercritical branching Markov processes

In this paper we establish spatial central limit theorems for a large class of supercritical branching Markov processes with general spatial-dependent branching mechanisms. These are generalizations of the spatial central limit theorems proved in [1] for branching OU processes with binary branching mechanisms. Compared with the results of [1], our central limit theorems are more satisfactory in the sense that the normal random variables in our theorems are non-degenerate.     

Almost sure central limit theorems for random functions

Let {Xn,-∞< n <∞} be a sequence of independent identically distributed random variables with EX1 = 0, EX12 = 1 and let Sn =∑k=1∞Xk, and Tn = Tn(X1,…,Xn) be a random function such that Tn = ASn Rn, where supn E|Rn| <∞and Rn = o(n~(1/2)) a.s., or Rn = O(n1/2-2γ) a.s., 0 <γ< 1/8. In this paper, we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure central limit theorem (FASCLT) for the random function Tn. As a consequence, it can be shown that ASCLT and FASCLT also hold for U-statistics, Von-Mises statistics, linear processes, moving average processes, error variance estimates in linear models, power sums, product-limit estimators of a continuous distribution, product-limit estimators of a quantile function, etc.     

A central limit theorem for random sums of random variables

Anscombe (1952) (also see Chung (1974)) has developed a central limit theoremof random sums of independent and identically distributed random variables. Applicability of this theorem in practice, however, is limited since the normalization requires random factors. In this paper we establish sufficient conditions under which the central limit theorem holds when such random factors are replaced by the underlying asymptotic mean and standard ddeviation. An application of this result in the context of shock models is also given.     

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We establish a quenched central limit theorem (CLT) for the branching Brownian motion with random immigration in dimension $d\geq4$. The limit is a Gaussian random measure, which is the same as the annealed central limit theorem, but the covariance kernel of the limit is different from that in the annealed sense when d=4.     

Systèmes quasi-orthogonaux et théorème central limite presque sur

We prove, in the case of irrational rotations, the existence of a generalized moment version of the Almost Sure Central Limit Theorem (ASCLT), with a speed of convergence. In our strategy, we use the notion of quasi-orthogonal system, and purpose a Gaussian randomization technic, new at least in this context, which should also permit to be able to treat problems of comparable nature.     

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本文在{ξi}为强混合样本,{ani}是实三角阵列下,得到了一个新的关于线性和n∑i=1aniξi的中心极限定理.并利用该中心极限定理,进一步建立了线性过程部分和的中心极限定理.     

随机环境中带移民的上临界分枝过程的Berry-Esseen界

考虑独立同分布的随机环境中带移民的上临界分枝过程(Zn).应用(Zn)与随机环境中不带移民分枝过程的联系,以及与相应随机游动的联系,在一些适当的矩条件下,本文证明关于log Zn的中心极限定理的Berry-Esseen界.     

Limit theorems in the Fourier transform method for the estimation of multivariate volatility

In this paper, we prove some limit theorems for the Fourier estimator of multivariate volatility proposed by Malliavin and Mancino (2002, 2009) [14] and [15]. In a general framework of discrete time observations we establish the convergence of the estimator and some associated central limit theorems with explicit asymptotic variance. In particular, our results show that this estimator is consistent for synchronous data, but possibly biased for non-synchronous observations. Moreover, from our general central limit theorem, we deduce that the estimator can be efficient in the case of a synchronous regular sampling. In the non-synchronous sampling case, the expression of the asymptotic variance is in general less tractable. We study this case more precisely through the example of an alternate sampling.     

A central limit theorem for Gibbs measures relative to Brownian motion

We study a Gibbs measure over Brownian motion with a pair potential which depends only on the increments. Assuming a particular form of this pair potential, we establish that in the infinite volume limit the Gibbs measure can be viewed as Brownian motion moving in a dynamic random environment. Thereby we are in a position to use the technique of Kipnis and Varadhan and to prove a functional central limit theorem.     

Inference for 2-D GARCH models

The purpose of this paper is, in the first step, to consider a class of GMM estimators with interesting asymptotic properties and a reasonable number of computations for two dimensionally indexed Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model. In the second step, we use the central limit theorem of Huang (1992) for spatial martingale differences to establish the LAN property for general two-dimensional discrete models on a regular grid with Gaussian errors. We then apply this result to the spatial GARCH model and derive the limit distribution of the maximum likelihood estimators of the parameters. Results of numerical simulations are presented.     

On the Central Paths in Symmetric Cone Programming

This paper is devoted to the study of optimal solutions of symmetric cone programs by means of the asymptotic behavior of central paths with respect to a broad class of barrier functions. This class is, for instance, larger than that typically found in the literature for semidefinite positive programming. In this general framework, we prove the existence and the convergence of primal, dual and primal–dual central paths. We are then able to establish concrete characterizations of the limit points of these central paths for specific subclasses. Indeed, for the class of barrier functions defined at the origin, we prove that the limit point of a primal central path minimizes the corresponding barrier function over the solution set of the studied symmetric cone program. In addition, we show that the limit points of the primal and dual central paths lie in the relative interior of the primal and dual solution sets for the case of the logarithm and modified logarithm barriers.