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.     

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.     

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 Random Fields of Negatively Associated Processes

A central limit theorem for negatively associated random fields is established under the fairly general conditions. We use the finite second moment condition instead of the finite (2+)th moment condition used by Roussas.⁽¹⁵⁾ A similar result is also given for positively associated sequences.     

A Functional Central Limit Theorem for Asymptotically Negatively Dependent Random Fields

Let X _k ; k N ^d be a random field which is asymptotically negative dependent in a certain sense. Define the partial sum process in the usual way so that , where . Under some suitable conditions, we show that W _n (·) converges in distribution to a Brownian sheet. Direct consequences of the result are functional central limit theorems for negative dependent random fields. The result is based on some general theorems concerning asymptotically negative dependent random fields, which are of independent interest.     

A Quantitative Central Limit Theorem for Linear Statistics of Random Matrix Eigenvalues

It is known that the fluctuations of suitable linear statistics of Haar distributed elements of the compact classical groups satisfy a central limit theorem. We show that if the corresponding test functions are sufficiently smooth, a rate of convergence of order almost 1/n can be obtained using a quantitative multivariate CLT for traces of powers that was recently proven using Stein’s method of exchangeable pairs.     

Quasi-sure Limit Theorem of Parabolic Stochastic Partial Differential Equations

In this paper we prove a quasi-sure limit theorem of parabolic stochastic partial differential equations with smooth coefficients and some initial conditions, by the way, we obtain the quasi-sure continuity of the solution.     

Quasi-sure Limit Theorem of Parabolic Stochastic Partial Differential Equations

In this paper we prove a quasi-sure limit theorem of parabolic stochastic partial differentialequations with smooth coefficients and some initial conditions,by the way,we obtain the quasi-surecontinuity of the solution.     

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.     

Statistical Version of the Central Limit Theorem for Vector-Valued Random Fields

The classical central limit theorem due to Newman for real-valued strictly stationary associated random fields is generalized to strictly stationary quasi-associated vector-valued random fields comprising, in particular, positively or negatively associated fields with finite second moments. We also establish a version of the CLT with random matrix normalization which allows us to construct approximate confidence intervals for the unknown mean vector.     

Central Limit Theorem for Linear Eigenvalue Statistics of Non-Hermitian Random Matrices

We consider large non-Hermitian random matrices X with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [72], or the distribution of the matrix elements needed to be Gaussian [73], or at least match the Gaussian up to the first four moments [82, 56]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of X with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices X that are presented in the companion paper [32]. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Central Limit Theorem for Linear Eigenvalue Statistics of Elliptic Random Matrices

We consider a class of elliptic random matrices which generalize two classical ensembles from random matrix theory: Wigner matrices and random matrices with iid entries. In particular, we establish a central limit theorem for linear eigenvalue statistics of real elliptic random matrices under the assumption that the test functions are analytic. As a corollary, we extend the results of Rider and Silverstein (Ann Probab 34(6):2118–2143, 2006) to real iid random matrices.     

任意随机变量序列的一个强极限定理

研究任意随机变量序列的强收敛性.利用鞅差序列级数收敛定理,证明了任意随机序列的一个强极限定理,作为推论,得到了马氏过程、鞅差序列及独立随机变量序列的强大数定律.     

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.     

关于两两NQD随机序列的一个极限定理

设为两两NQD随机序列, 且, 是一列严格单调递增的凸序列. 本文将 Feller (1946)关于独立同分布期望不存在随机序列的极限定理推广到两两NQD随机 序列的情形.     

NA及LNQD随机变量列的几乎处处中心极限定理

本文在二阶矩存在的条件下,证明了NA及LNQD随机变量列的几乎处处中心极限定理,使主要结果成立,其中W为[0,1]上标准Brown运动。     

Rates in the Empirical Central Limit Theorem for Stationary Weakly Dependent Random Fields

A weak dependence condition is derived as the natural generalization to random fields on notions developed in Doukhan and Louhichi (1999). Examples of such weakly dependent fields are defined. In the context of a weak dependence coefficient series with arithmetic or geometric decay, we give explicit bounds in Prohorov metric for the convergence in the empirical central limit theorem. For random fields indexed by &Zopf ^d , in the geometric decay case, rates have the form n ^−1/(8d+24) L(n), where L(n) is a power of log(n). This revised version was published online in June 2006 with corrections to the Cover Date.