CLT for linear random fields with stationary martingale-difference innovation

In [V. Paulauskas, On Beveridge–Nelson decomposition and limit theorems for linear random fields, J. Multivariate Anal., 101:621–639, 2010], limit theorems for linear random fields generated by independent identically distributed innovations were proved. In this paper, we present the central limit theorem for linear random fields with martingale-differences innovations satisfying the central limit theorem from [J. Dedecker, A central limit theorem for stationary random fields, Probab. Theory Relat. Fields, 110(3):397–426, 1998] and arranged in lexicographical order.     

A Conditional CLT which Fails for Ergodic Components

We show that the conditional central limit theorem can take place for a stationary process defined on a nonergodic dynamical system while this last does not satisfy the central limit theorem for any ergodic component. There exists an ergodic Markov chain such that the conditional central limit theorem is satisfied for an invariant measure but fails to hold for almost all starting points.      

A CLT for renewal processes with a finite set of interarrival distributions

We prove a central limit theorem for a renewal process based on a sequence of independent non-negative interarrival times whose distributions are taken from a finite set. The result extends the classical central limit theorem obtained by Takács (1956).     

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.     

Central Limit Theorems for Uniform Model Random Polygons

We show how a central limit theorem for Poisson model random polygons implies a central limit theorem for uniform model random polygons. To prove this implication, it suffices to show that in the two models, the variables in question have asymptotically the same expectation and variance. We use integral geometric expressions for these expectations and variances to reduce the desired estimates to the convergence $(1+\frac{\alpha}{n})^{n}\to e^{\alpha}$ as n????.     

Limit theorems for Markov chains of finite rank

We consider a Markov chain with a general state space, but whose behavior is governed by finite matrices. After a brief exposition of the basic properties of this chain, its convenience as a model is illustrated by three limit theorems. The ergodic theorem, the central limit theorem, and an extreme-value theorem are expressed in terms of dominant eigenvalues of finite matrices and proved by simple matrix theory.     

Central Limit Theorems for Asymptotically Negatively Associated Random Fields

Abstract The aim of this paper is to investigate the central limit theorems for asymptotically negatively dependent random fields under lower moment conditions or the Lindeberg condition. Results obtained improve a central limit theorem of Roussas [11] for negatively assiated fields and the main results of Su and Chi [18], and also include a central limit of theorem for weakly negatively associated random variables similar to that of Burton et al. [20]. Research supported by National Natural Science Foundation of China (No. 19701011)     

Statistical properties for nonhyperbolic maps with finite range structure

We establish the central limit theorem and non-central limit theorems for maps admitting indifferent periodic points (the so-called intermittent maps). We also give a large class of Darling-Kac sets for intermittent maps admitting infinite invariant measures. The essential issue for the central limit theorem is to clarify the speed of -convergence of iterated Perron-Frobenius operators for multi-dimensional maps which satisfy Renyi's condition but fail to satisfy the uniformly expanding property. Multi-dimensional intermittent maps typically admit such derived systems. There are examples in section 4 to which previous results on the central limit theorem are not applicable, but our extended central limit theorem does apply.

     

A functional limit theorem for <Emphasis Type="Italic">η</Emphasis>-weakly dependent processes and its applications

We prove a general functional central limit theorem for weak dependent time series. A very large variety of models, for instance, causal or non causal linear, ARCH(∞), LARCH(∞), Volterra processes, satisfies this theorem. Moreover, it provides numerous applications as well for bounding the distance between the empirical mean and the Gaussian measure than for obtaining central limit theorem for sample moments and cumulants. C. José Rafael León—Partially supported by the program ECOS-NORD of Fonacit, Venezuela.     

A universal result in almost sure central limit theory

The discovery of the almost sure central limit theorem (Brosamler, Math. Proc. Cambridge Philos. Soc. 104 (1988) 561–574; Schatte, Math. Nachr. 137 (1988) 249–256) revealed a new phenomenon in classical central limit theory and has led to an extensive literature in the past decade. In particular, a.s. central limit theorems and various related ‘logarithmic’ limit theorems have been obtained for several classes of independent and dependent random variables. In this paper we extend this theory and show that not only the central limit theorem, but every weak limit theorem for independent random variables, subject to minor technical conditions, has an analogous almost sure version. For many classical limit theorems this involves logarithmic averaging, as in the case of the CLT, but we need radically different averaging processes for ‘more sensitive’ limit theorems. Several examples of such a.s. limit theorems are discussed.     

Invariance principles under weak dependence

The aim of this paper is to give a functional form for the central limit theorem obtained by Bradley for strong mxing sequences of random variables, under a certain assumption about the size of the maximal coefficients of correlations. The convergence of the moments of order 2 + δ in the central limit theorem for this class of random variables is also obtained.     

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.     

Homogenization of random transport along periodic two-dimensional flows

We present a model for random transport along periodic two-dimensional flows and use the concept of rotation numbers from dynamical systems to prove a functional central limit theorem for this model. The limiting law turns out to be a stable Lévy process.     

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.     

Multidimensional compound Poisson distributions in free probability

Inspired by Speicher's multidimensional free central limit theorem and semicircle families, we prove an in?nite dimensional compound Poisson limit theorem in free probability, and de?ne in?nite dimensional compound free Poisson distributions in a non-commutative probability space. In?nite dimensional free in?nitely divisible distributions are de?ned and characterized in terms of their free cumulants. It is proved that for a sequence of random variables, the following three statements are equivalent:(1) the distribution of the sequence is multidimensional free in?nitely divisible;(2) the sequence is the limit in distribution of a sequence of triangular trays of families of random variables;(3) the sequence has the same distribution as that of {a_1~((i)): i = 1, 2,...}of a multidimensional free L′evy process {{a_1~((i)): i = 1, 2,...} : t≥0}. Under certain technical assumptions, this is the case if and only if the sequence is the limit in distribution of a sequence of sequences of random variables having multidimensional compound free Poisson distributions.     

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.     

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.     

Analysis of a generalized Friedman’s urn with multiple drawings

We study a generalized Friedman’s urn model with multiple drawings of white and blue balls. After a drawing, the replacement follows a policy of opposite reinforcement. We give the exact expected value and variance of the number of white balls after a number of draws, and determine the structure of the moments. Moreover, we obtain a strong law of large numbers, and a central limit theorem for the number of white balls. Interestingly, the central limit theorem is obtained combinatorially via the method of moments and probabilistically via martingales. We briefly discuss the merits of each approach. The connection to a few other related urn models is briefly sketched.     

Universality for the Conjugate Gradient and MINRES Algorithms on Sample Covariance Matrices

We present a probabilistic analysis of two Krylov subspace methods for solving linear systems. We prove a central limit theorem for norms of the residual vectors that are produced by the conjugate gradient and MINRES algorithms when applied to a wide class of sample covariance matrices satisfying some standard moment conditions. The proof involves establishing a four-moment theorem for the so-called spectral measure, implying, in particular, universality for the matrix produced by the Lanczos iteration. The central limit theorem then implies an almost-deterministic iteration count for the iterative methods in question. © 2022 Wiley Periodicals LLC.     

A tracial quantum central limit theorem

We prove a central limit theorem for non-commutative random variables in a von Neumann algebra with a tracial state: Any non-commutative polynomial of averages of i.i.d. samples converges to a classical limit. The proof is based on a central limit theorem for ordered joint distributions together with a commutator estimate related to the Baker-Campbell-Hausdorff expansion. The result can be considered a generalization of Johansson's theorem on the limiting distribution of the shape of a random word in a fixed alphabet as its length goes to infinity.