A Central Limit Theorem for isotropic flows

We establish that the image of a measure, which satisfies a certain energy condition, moving under a standard isotropic Brownian flow will, when properly scaled, have an asymptotically normal distribution under almost every realization of the flow. We derive the same result for an initial point mass moved by an isotropic Kraichnan flow.     

A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue

We consider a model in which the production of new molecules in a chemical reaction network occurs in a seemingly stochastic fashion, and can be modeled as a Poisson process with a varying arrival rate: the rate is λ _i when an external Markov process J(?) is in state i. It is assumed that molecules decay after an exponential time with mean μ ^?1. The goal of this work is to analyze the distributional properties of the number of molecules in the system, under a specific time-scaling. In this scaling, the background process is sped up by a factor N ^α , for some α>0, whereas the arrival rates become N λ _i , for N large. The main result of this paper is a functional central limit theorem (F-CLT) for the number of molecules, in that, after centering and scaling, it converges to an Ornstein-Uhlenbeck process. An interesting dichotomy is observed: (i) if α > 1 the background process jumps faster than the arrival process, and consequently the arrival process behaves essentially as a (homogeneous) Poisson process, so that the scaling in the F-CLT is the usual \(\sqrt {N}\), whereas (ii) for α≤1 the background process is relatively slow, and the scaling in the F-CLT is N ^1?α/2. In the latter regime, the parameters of the limiting Ornstein-Uhlenbeck process contain the deviation matrix associated with the background process J(?).     

A Central Limit Theorem for linear random fields

A Central Limit Theorem is proved for linear random fields when sums are taken over union of finitely many disjoint rectangles. The approach does not rely upon the use of Beveridge-Nelson decomposition and the conditions needed are similar in nature to those given by Ibragimov for linear processes. When specializing this result to the case when sums are being taken over rectangles, a complete analogue of the Ibragimov result is obtained for random fields with a lot of uniformity.     

A Mean Central Limit Theorem for Multiplicative Systems

The present paper deals with an error-estimate in the mean central limit theorem for strongly multiplicative systems. The appearing absolute constant in the error-bound is computed.     

A Counterexample in the Central Limit Theorem

We construct a c₀-valued random variable X such that (S_n/n)_nNhas a convergent subsequence, but X does not satisfy the centrallimit theorem. 1991 Mathematics Subject Classification 60B12.     

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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.     

A Central Limit Theorem for a Localized Version of the SK Model

In this note, we consider a SK (Sherrington–Kirkpatrick)-type model on ℤ ^d for d≥1, weighted by a function allowing to any single spin to interact with a small proportion of the other ones. In the thermodynamical limit, we investigate the equivalence of this model with the usual SK spin system, through the study of the fluctuations of the free energy. This author’s research partially supported by CAPES.     

Central Limit Theorem for Random Strict Partitions

We consider the set of all partitions of a number n into distinct summands (the so-called strict partitions) with the uniform distribution on it and study fluctuations of a random partition near its limit shape, for large n. The use of geometrical language allows us to state the problem in terms of the limit behavior of random step functions (Young diagrams). A central limit theorem for such functions is proven. Our method essentially uses the notion of large canonical ensemble of partitions. Bibliography: 7 titles.     

Spatial Central Limit Theorem for Supercritical Superprocesses

We consider a measure-valued diffusion (i.e., a superprocess). It is determined by a couple \((L,\psi )\), where L is the infinitesimal generator of a strongly recurrent diffusion in \(\mathbb {R}^{d}\) and \(\psi \) is a branching mechanism assumed to be supercritical. Such processes are known, see for example, (Englander and Winter in Ann Inst Henri Poincaré 42(2):171–185, 2006), to fulfill a law of large numbers for the spatial distribution of the mass. In this paper, we prove the corresponding central limit theorem. The limit and the CLT normalization fall into three qualitatively different classes arising from “competition” of the local growth induced by branching and global smoothing due to the strong recurrence of L. We also prove that the spatial fluctuations are asymptotically independent of the fluctuations of the total mass of the process.     

A Central Limit Theorem for Non-stationary Strongly Mixing Random Fields

In this paper we extend a central limit theorem of Peligrad for uniformly strong mixing random fields satisfying the Lindeberg condition in the absence of stationarity property. More precisely, we study the asymptotic normality of the partial sums of uniformly \(\alpha \)-mixing non-stationary random fields satisfying the Lindeberg condition, in the presence of an extra dependence assumption involving maximal correlations.     

A Central Limit Theorem for Convex Chains in the Square

Points P ₁ ,... ,P _n in the unit square define a convex n -chain if they are below y=x and, together with P ₀ =(0,0) and P _n+1 =(1,1) , they are in convex position. Under uniform probability, we prove an almost sure limit theorem for these chains that uses only probabilistic arguments, and which strengthens similar limit shape statements established by other authors. An interesting feature is that the limit shape is a direct consequence of the method. The main result is an accompanying central limit theorem for these chains. A weak convergence result implies several other statements concerning the deviations between random convex chains and their limit. Received April 17, 1998, and in revised form December 4, 1998.     

A Functional Central Limit Theorem for Positively Dependent Random Variables

In this note we prove a functional central limit theorem for LPQD processes, satisfying some assumptions on the covariances and the moment condition sup_j≥1E|X_j|^2+ρ < ∞ for some ρ > 0.     

中心极限定理教学中的一个问题

指出并分析了目前国内概率论教材中中心极限定理部分存在的一个问题.     

Central Limit Theorem for a Class of Nonhomogeneous Random Walks

A spatially nonhomogeneous random walk _t on the grid =^m X ⁿ is considered. Let _t ⁰ be a random walk homogeneous in time and space, and let _t be obtained from it by changing transition probabilities on the set A= X ⁿ, || < , so that the walk remains homogeneous only with respect to the subgroup ⁿ of the group . It is shown that if >m 2 or the drift is distinct from zero, then the central limit theorem holds for _t.     

A Maximal Inequality and a Functional Central Limit Theorem for set-indexed empirical processes

For the tail probabilities of a general set-indexed empirical process in an arbitrary sample space a maximal inequality is derived. In the case that the class of sets by which the process is indexed possesses a total ordering, the application of our inequality yields an elementary proof for a functional central limit theorem without involving such advanced techniques as symmetrization, stratification, chaining or Gaussian domination. Analogously, the inequality leads to a weak uniform law of large numbers (including convergence rate).     

The Central Limit Theorem for Free Additive Convolution

Let {X_n}^∞_n=1be a sequence of free, identically distributed random variables with common distributionμ. Then there exist sequences {B_n}^∞_n=1and {A_n}^∞_n=1of positive and real numbers, respectively, such that sequence of random variables[formula]converges in distribution to the semicircle law if and only if the function[formula]is slowly varying in Karamata's sense. In other words, the free domain of attraction of the semicircle law coincides with the classical domain of attraction of the Gaussian. We prove an analogous result for normal domains of attraction in the sense of Linnik.     

The Central Limit Theorem for Nonhomogeneous Markov Chains

In this paper, we prove a new central limit theorem for nonhomogeneous Markov chain by using the martingale central limit theorem under the condition of convergence of transition probability matrices for nonhomogeneous Markov chain in Cesaro sense, which can not be implied by Dobrushin's work.