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.     

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.     

Expansions for the Distributions of Some Normalized Summations of Random Numbers of I.I.D. Random Variables

The central limit theorem for a normalized summation of random number of i.i.d. random variables is well known. In this paper we improve the central limit theorem by providing a two-term expansion for the distribution when the random number is the first time that a simple random walk exceeds a given level. Some numerical evidences are provided to show that this expansion is more accurate than the simple normality approximation for a specific problem considered.     

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We establish a quenched central limit theorem (CLT) for the branching Brownian motion with random immigration in dimension $d\geq4$. The limit is a Gaussian random measure, which is the same as the annealed central limit theorem, but the covariance kernel of the limit is different from that in the annealed sense when d=4.     

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.

     

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.     

An almost sure central limit theorem for the weight function sequences of NA random variables

Consider the weight function sequences of NA random variables. This paper proves that the almost sure central limit theorem holds for the weight function sequences of NA random variables. Our results generalize and improve those on the almost sure central limit theorem previously obtained from the i.i.d. case to NA sequences.     

Derived random measures

A derived random measure is constructed by integration of a random process with respect to a random measure independent of that process. Basic distributional properties, a continuity theorem, sample path properties, a strong law of large numbers, and a central limit theorem for derived random measures are established. Applications are given to compounding and thinning of point processes and the measure of a random set.     

Stein's method and random character ratios

Stein's method is used to prove limit theorems for random character ratios. Tools are developed for four types of structures: finite groups, Gelfand pairs, twisted Gelfand pairs, and association schemes. As one example an error term is obtained for a central limit theorem of Kerov on the spectrum of the Cayley graph of the symmetric group generated by -cycles. Other main examples include an error term for a central limit theorem of Ivanov on character ratios of random projective representations of the symmetric group, and a new central limit theorem for the spectrum of certain random walks on perfect matchings. The results are obtained with very little information: a character formula for a single representation close to the trivial representation and estimates on two step transition probabilities of a random walk. The limit theorems stated in this paper are for normal approximation, but many of the tools developed are applicable for arbitrary distributional approximation.

     

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.     

Limit behaviors of random connected graphs driven by a Poisson process

We consider a class of random connected graphs with random vertices and random edges with the random distribution of vertices given by a Poisson point process with the intensity n localized at the vertices and the random distribution of the edges given by a connection function. Using the Avram-Bertsimas method constructed in 1992 for the central limit theorem on Euclidean functionals, we find the convergence rate of the central limit theorem process, the moderate deviation, and an upper bound for large deviations depending on the total length of all edges of the random connected graph.     

The characteristic polynomial of a random permutation matrix

We establish a central limit theorem for the logarithm of the characteristic polynomial of a random permutation matrix. We relate this result to a central limit theorem of Wieand for the counting function for the eigenvalues lying in some interval on the unit circle.     

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.     

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)     

Central limit theorem for linear processes generated by IID random variables under the sub-linear expectation

In this paper, we investigate the central limit theorem and the invariance principle for linear processes generated by a new notion of independently and identically distributed(IID) random variables for sub-linear expectations initiated by Peng [19]. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov's central limit theorem and invariance principle to the case where probability measures are no longer additive.     

次线性期望下的一般中心极限定理

受Peng-中心极限定理的启发,本文主要应用G-正态分布的概念,放宽Peng-中心极限定理的条件,在次线性期望下得到形式更为一般的中心极限定理.首先,将均值条件E[X_n]=ε[X_n]=0放宽为|E[X_n]|+|ε[X_n]|=O(1/n);其次,应用随机变量截断的方法,放宽随机变量的2阶矩与2+δ阶矩条件;最后,将该定理的Peng-独立性条件进行放宽,得到卷积独立随机变量的中心极限定理.     

A central limit theorem for exchangeable random variables on a locally compact abelian group

A central limit theorem is given for uniformly infinitesimal triangular arrays of random variables in which the random variables in each row are exchangeable and take values in a locally compact second countable abelian group. The limiting distribution in the theorem is Gaussian.     

On the convergence of vector random measures

Summary The aim of this paper is to study Banach space-valued symmetric independently scattered random measures with emphasis on their convergence properties. The Vitali-Hahn-Saks Theorem, the Skorokhod theorem about the relations between the convergence a.e. and the convergence in law of random variables, and the central limit theorem for Banach valued random variables due to Hoffmann-Jorgensen, Pisier are extended to such measures.     

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.     

The central limit theorem for a sequence of random processes with space-varying long memory*

In this paper, we investigate a sequence of square-integrable random processes with space-varying memory. We establish sufficient conditions for the central limit theorem in the space L ²(μ) for the partial sums of the sequence of random processes with space-varying long memory. Of particular interest is a nonstandard normalization of the partial sums in the central limit theorem.