A central limit theorem for mixing stationary point processes

Suppose A₀ is a strictly stationary, second order point process on Z^d that is ?-mixing. The particles initially present are then continually subjected to random translations via random walks. If A_n is the point process resulting at time n, then we prove, under certain technical conditions, that the total occupation time by time n of a finite nonempty subset B of Z^d, namely, S_n(B)=Σⁿ_k=1A_k(B), is asymptotically normally distributed.     

Self-normalized central limit theorem for sums of weakly dependent random variables

Let {X _n,n1} be a strictly stationary sequence of weakly dependent random variables satisfyingEX _n=,EX _n ² <,Var S _n /n² and the central limit theorem. This paper presents two estimators of ². Their weak and strong consistence as well as their rate of convergence are obtained for -mixing, -mixing and associated sequences.Supported by a NSF grant and a Taft travel grant. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025.Supported by a Taft Post-doctoral Fellowship at the University of Cincinnati and by the Fok Yingtung Education Foundation of China. Hangzhou University, Hangzhou, Zhejiang, P.R. China and Department of Mathematics, National University of Singapore, Singapore 0511.     

An elementary proof of the local central limit theorem

We give an elementary proof of the local central limit theorem for independent, non-identically distributed, integer valued and vector valued random variables.Support by a NSF Grant.Supported by an NSERC Grant     

Central limit theorem and almost sure central limit theorem for the product of some partial sums

In this paper, we give the central limit theorem and almost sure central limit theorem for products of some partial sums of independent identically distributed random variables.     

On Kersting's proof of the central limit theorem

By an adaptation of a method originally invented by G. Kersting [1] for the calculation of the limiting distribution of Markovian processes the central limit theorem (CLT) is proven. Only the case of equal variances is considered.     

Simple transient random walks in one-dimensional random environment: the central limit theorem

We consider a simple random walk (dimension one, nearest neighbour jumps) in a quenched random environment. The goal of this work is to provide sufficient conditions, stated in terms of properties of the environment, under which the central limit theorem (CLT) holds for the position of the walk. Verifying these conditions leads to a complete solution of the problem in the case of independent identically distributed environments as well as in the case of uniformly ergodic (and thus also weakly mixing) environments.      

On the almost sure (a.s.) central limit theorem for random variables with infinite variance

We give criteria for a sequence (X _n ) of i.i.d.r.v.'s to satisfy the a.s. central limit theorem, i.e.,

Power-law estimates for the central limit theorem for convex sets

We investigate the rate of convergence in the central limit theorem for convex sets established in [B. Klartag, A central limit theorem for convex sets, Invent. Math., in press. [8]]. We obtain bounds with a power-law dependence on the dimension. These bounds are asymptotically better than the logarithmic estimates which follow from the original proof of the central limit theorem for convex sets.     

A note on applications of the martingale central limit theorem to random permutations

A unified martingale approach is presented for establishing the asymptotic normality of some sequences of random variables. It is applied to the numbers of inversions, rises, and peaks, respectively, as well as the oscillation and the sum of consecutive pair products of a random permutation. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10, 323–332 (1997)     

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.     

On the central limit theorem for modulus trimmed sums

We prove a functional central limit theorem for modulus trimmed i.i.d. variables in the domain of attraction of a nonnormal stable law. In contrast to the corresponding result under ordinary trimming, our CLT contains a random centering factor which is inevitable in the nonsymmetric case. The proof is based on the weak convergence of a two-parameter process where one of the parameters is time and the second one is the fraction of truncation.     

A general central limit theorem for strong mixing sequences

A central limit theorem for strong mixing sequences is given that applies to both non-stationary sequences and triangular array settings. The result improves on an earlier central limit theorem for this type of dependence given by Politis, Romano and Wolf in 1997.     

On the central limit theorem for the stationary Poisson process of compact sets

Stochastic geometry models based on a stationary Poisson point process of compact subsets of the Euclidean space are examined. Random measures on ?^d, derived from these processes using Hausdorff and projection measures are studied. The central limit theorem is formulated in a way which enables comparison of the various estimators of the intensity of the produced random measures. Approximate confidence intervals for the intensity are constructed. Their use is demonstrated in an example of length intensity estimation for the segment processes. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Refinement of the almost sure central limit theorem for associated processes

In this paper, for the partial sumsS _n of a stationary associated random process it is proved that the logarithmic averages converge almost surely. The asymptotic normality of the normalized difference between the logarithmic averages and their limiting value is established. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 513–522, October, 2000.     

A central limit theorem for branching Brownian motion with random immigration

We establish a central limit theorem for a branching Brownian motion with random immigration under the annealed law,where the immigration is determined by another branching Brownian motion.The limit is a Gaussian random measure and the normalization is t3/4for d=3 and t1/2for d≥4,where in the critical dimension d=4 both the immigration and the branching Brownian motion itself make contributions to the covariance of the limit.     

Covariance kernel and the central limit theorem in the total variation distance

We modify and generalize the idea of covariance kernels for Borel probability measures on R^d, and study the relation between the central limit theorem in the total variation distance and the convergence of covariance kernels.     

The central limit theorem for time series regression

The central limit problem is considered for a simple regression, where the residuals, x(n), are stationary and the sequence regressed on y^(N)(n), may depend on the number of observations, N, to hand. Two situations are considered, one where the residual is generated by a linear process (i.e. the best linear predictor is the best predictor) and the more general situation where that is not so. Two types of conditions are needed, the first of which limits the contribution of any individual y^(N)(n) and the second of which relates to the mixing properties of x(n). If ε(n) is the linear innovation sequence, in the linear case, with $\text{lim}{}_{\mathrm{k}}\text{→∞}\text{E}\text{(ε(n)}{}^2\text{∥}\text{F}{}_{\mathrm{n?k}}\text{)=}\text{F}{}^2\text{,}\text{F}{}_{\mathrm{n}}$ being the associated family of o-algebra, then the central limit theorem holds under minimal conditions on y^(N)(n). Under sligthly stronger conditions on y^(N)(n) and for x(n) weakly mixing this theorem and associated theorems, are shown to hold under further fairly weak conditions on the dependence of x(n) on its past.     

ON THE CENTRAL LIMIT THEOREM IN PRODUCT SPACES

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