多维随机向量序列加权和的中心极限定理

本文研究了多维随机向量序列加权和的渐近行为.利用Lindeberg中心极限定理的基本思想,得到了多维随机向量序列加权和的中心极限定理及其收敛速度,为Lindeberg中心极限定理的推广.     

Asymptotic normality for two-stage sampling from a finite population

Summary In this paper we consider two-stage sampling from a finite population, and associated estimators of the population total, in a general setting which includes most two-stage procedures in the literature. The main result gives general conditions for asymptotic normality of the estimators. The proof is based on a martingale central limit theorem. It is indicated how the result can be extended to multi-stage procedures.     

The class of tenable zero-balanced Pólya urns with an initially dominant subset of colors

In Kholfi and Mahmoud (2011) the class of tenable irreducible nondegenerate zero-balanced Pólya urn schemes is introduced and its asymptotic behavior in various phases is studied. In the absence of an initially dominant subset of colors, the counts of balls of all the colors satisfy multivariate central limit theorems. It is reported there that the case of an initially dominant subset of colors poses challenges requiring finer asymptotic analysis. In the present investigation we follow up on this. Indeed, we characterize noncritical cases with an initially dominant subset of colors in which not all ball counts satisfy one multivariate central limit theorem, but rather a subset of the ball counts satisfies a singular multivariate central limit theorem. The rest of the cases are critical, in which all the ball counts satisfy a multivariate central limit theorem, but under a different scaling. However, for these critical cases the Gaussian phases are delayed considerably.     

On Asymptotic Expansion in the Random Allocation of Particles by Sets

We consider a scheme of equiprobable allocation of particles into cells by sets. An Edgeworth-type asymptotic expansion in the local central limit theorem for the number of empty cells left after allocation of all sets of particles is derived.     

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.     

Self-crossing Points of a Line Segment Process

This paper is devoted to planar stationary line segment processes. The segments are assumed to be independent, identically distributed, and independent of the locations (reference points). We consider a point process formed by self-crossing points between the line segments. Its asymptotic variance is explicitly expressed for Poisson segment processes. The main result of the paper is the central limit theorem for the number of intersection points in expanding rectangular sampling window. It holds not only for Poisson processes of reference points but also for stationary point processes satisfying certain conditions on absolute regularity (β-mixing) coefficients. The proof is based on the central limit theorem for β-mixing random fields. Approximate confidence intervals for the intensity of intersections can be constructed.     

An estimator for the quadratic covariation of asynchronously observed Itô processes with noise: Asymptotic distribution theory

The article is devoted to the nonparametric estimation of the quadratic covariation of non-synchronously observed Itô processes in an additive microstructure noise model. In a high-frequency setting, we aim at establishing an asymptotic distribution theory for a generalized multiscale estimator including a feasible central limit theorem with optimal convergence rate on convenient regularity assumptions. The inevitably remaining impact of asynchronous deterministic sampling schemes and noise corruption on the asymptotic distribution is precisely elucidated. A case study for various important examples, several generalizations of the model and an algorithm for the implementation warrant the utility of the estimation method in applications.     

Approximate Central Limit Theorems

We refine the classical Lindeberg–Feller central limit theorem by obtaining asymptotic bounds on the Kolmogorov distance, the Wasserstein distance, and the parameterized Prokhorov distances in terms of a Lindeberg index. We thus obtain more general approximate central limit theorems, which roughly state that the row-wise sums of a triangular array are approximately asymptotically normal if the array approximately satisfies Lindeberg’s condition. This allows us to continue to provide information in nonstandard settings in which the classical central limit theorem fails to hold. Stein’s method plays a key role in the development of this theory.     

Asymptotic normality of spectral estimates

The asymptotic normality of some spectral estimates, including a functional central limit theorem for an estimate of the spectral distribution function, is proved for fourth-order stationary processes. In contrast to known results it is not assumed that all moments exist or that the process is linear. The data are allowed to be tapered. Using some recent results on the central limit theorem for stationary processes, corollaries are obtained for strong and φ-mixing sequences and linear transformations of martingale differences.     

Asymptotic Distribution of a Kind of Dirichlet Distribution

The Dirichlet distribution that we are concerned with in this paper is very special, in which all parameters are different from each other. We prove that the asymptotic distribution of this kind of Dirichlet distributions is a normal distribution by using the central limit theorem and Slutsky theorem.     

A functional central limit theorem for a generalized occupancy problem

A functional central limit theorem is proved for a class of finitely exchangeable random variables which are based on an occupancy scheme.     

An Asymptotic Sampling Recomposition Theorem for Gaussian Signals

For any Gaussian signal and every given sampling frequency we prove an asymptotic property of type Shannon’s sampling theorem, based on normalized cardinal sines, which keeps constant the sampling frequency. We generalize the Shannon’s sampling theorem for a class of non band–limited signals which plays a central role in the signal theory, the Gaussian map is the unique function which reachs the minimum of the product of the temporal and frecuential width. This solve a conjecture stated in [1] and suggested by [3].     

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.     

系统辨识中LS估计的渐近正态性的注记

文献[1]中,我们用有关鞅的中心极限定理,证明了系统辨识中LS估计的渐近正态性。然而[1]中的条件是苛刻的。本文利用Mcleish的相依变量的中心极限定理改进了[1]的结果。     

Central limit asymptotics for shifts of finite type

We study the rate of convergence and asymptotic expansions in the central limit theorem for the class of Hölder continuous functions on a shift of finite type endowed with a stationary equilibrium state. It is shown that the rate of convergence in the theorem isO(n ^?1/2) and when the function defines a non-lattice distribution an asymptotic expansion to the order ofo(n ^?1/2) is given. Higher-order expansions can be obtained for a subclass of functions. We also make a remark on the central limit theorem for (closed) orbital measures.     

The proportional hazards model for survey data from independent and clustered super-populations

Data from most complex surveys are subject to selection bias and clustering due to the sampling design. Results developed for a random sample from a super-population model may not apply. Ignoring the survey sampling weights may cause biased estimators and erroneous confidence intervals. In this paper, we use the design approach for fitting the proportional hazards (PH) model and prove formally the asymptotic normality of the sample maximum partial likelihood (SMPL) estimators under the PH model for both stochastically independent and clustered failure times. In the first case, we use the central limit theorem for martingales in the joint design-model space, and this enables us to obtain results for a general multistage sampling design under mild and easily verifiable conditions. In the case of clustered failure times, we require asymptotic normality in the sampling design space directly, and this holds for fewer sampling designs than in the first case. We also propose a variance estimator of the SMPL estimator. A key property of this variance estimator is that we do not have to specify the second-stage correlation model.     

A central limit theorem for extremal characters of the infinite symmetric group

The asymptotic behavior of the lengths of the first rows and columns in the random Young diagrams corresponding to extremal characters of the infinite symmetric group is studied. We consider rows and columns with linear growth in n and prove a central limit theorem for their lengths in the case of distinct Thoma parameters. We also prove a more precise statement relating the growth of rows and columns of Young diagrams to a simple independent random sampling model.     

Flexible sliced designs for computer experiments

Sliced Latin hypercube designs are popularly adopted for computer experiments with qualitative factors. Previous constructions require the sizes of different slices to be identical. Here we construct sliced designs with flexible sizes of slices. Besides achieving desirable one-dimensional uniformity, flexible sliced designs (FSDs) constructed in this paper accommodate arbitrary sizes for different slices and cover ordinary sliced Latin hypercube designs as special cases. The sampling properties of FSDs are derived and a central limit theorem is established. It shows that any linear combination of the sample means from different models on slices follows an asymptotic normal distribution. Some simulations compare FSDs with other sliced designs in collective evaluations of multiple computer models.     

Central limit theorem for Gibbsian U-statistics of facet processes

A special case of a Gibbsian facet process on a fixed window with a discrete orientation distribution and with increasing intensity of the underlying Poisson process is studied. All asymptotic joint moments for interaction U-statistics are calculated and the central limit theorem is derived using the method of moments.     

Particle-based online estimation of tangent filters with application to parameter estimation in nonlinear state-space models

This paper presents a novel algorithm for efficient online estimation of the filter derivatives in general hidden Markov models. The algorithm, which has a linear computational complexity and very limited memory requirements, is furnished with a number of convergence results, including a central limit theorem with an asymptotic variance that can be shown to be uniformly bounded in time. Using the proposed filter derivative estimator, we design a recursive maximum likelihood algorithm updating the parameters according the gradient of the one-step predictor log-likelihood. The efficiency of this online parameter estimation scheme is illustrated in a simulation study.