Conditioned rates of convergence in the CLT for sums and maximum sums

An interesting recent result of Landers and Roggé (1977, Ann. Probability5, 595–600) is investigated further. Rates of convergence in the conditioned central limit theorem are developed for partial sums and maximum partial sums, with positive mean and zero mean separately, of sequences of independent identically distributed random variables. As corollaries we obtain a conditioned central limit theorem for maximum partial sums both for positive and zero mean cases.     

Mixing normal approximations of vectors of sums and maximum sums

Summary The conditioned central limit theorem for the vector of maximum partial sums based on independent identically distributed random vectors is investigated and the rate of convergence is discussed. The conditioning is that of Rényi (1958,Acta Math. Acad. Sci. Hungar.,9, 215–228). Analogous results for the vector of partial sums are obtained. University of Petroleum and Minerals     

Complete convergence theorems for L-mixingales

Sufficient conditions for the complete convergence for the partial sums and the random selected partial sums of L^p-mixingales are given. Necessary conditions are also discussed.     

On the deviation between the distributions of sums and maximum sums of IID random vectors

Summary For a sequence of independent and identically distributed random vectors, with finite moment of order less than or equal to the second, the rate at which the deviation between the distribution functions of the vectors of partial sums and maximums of partial sums is obtained both when the sample size is fixed and when it is random, satisfying certain regularity conditions. When the second moments exist the rate is of ordern ^−1/4 (in the fixed sample size case). Two applications are given, first, we compliment some recent work of Ahmad (1979,J. Multivariate Anal.,9, 214–222) on rates of convergence for the vector of maximum sums and second, we obtain rates of convergence of the concentration functions of maximum sums for both the fixed and random sample size cases.     

Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications

In this paper, we establish some Rosenthal type inequalities for maximum partial sums of asymptotically almost negatively associated random variables, which extend the corresponding results for negatively associated random variables. As applications of these inequalities, by employing the notions of residual Cesàro α-integrability and strong residual Cesàro α-integrability, we derive some results on L _p convergence where 1 < p < 2 and complete convergence. In addition, we estimate the rate of convergence in Marcinkiewicz-Zygmund strong law for partial sums of identically distributed random variables.     

The central limit theorem for sums of trimmed variables with heavy tails

Trimming is a standard method to decrease the effect of large sample elements in statistical procedures, used, e.g., for constructing robust estimators and tests. Trimming also provides a profound insight into the partial sum behavior of i.i.d. sequences. There is a wide and nearly complete asymptotic theory of trimming, with one remarkable gap: no satisfactory criteria for the central limit theorem for modulus trimmed sums have been found, except for symmetric random variables. In this paper we investigate this problem in the case when the variables are in the domain of attraction of a stable law. Our results show that for modulus trimmed sums the validity of the central limit theorem depends sensitively on the behavior of the tail ratio P(X>t)/P(|X|>t) of the underlying variable X as t→∞ and paradoxically, increasing the number of trimmed elements does not generally improve partial sum behavior.     

The Moment Convergence Rates for Largest Eigenvalues of β Ensembles

The paper focuses on the largest eigenvalues of the β-Hermite ensemble and the β-Laguerre ensemble. In particular, we obtain the precise moment convergence rates of their largest eigenvalues. The results are motivated by the complete convergence for partial sums of i.i.d. random variables, and the proofs depend on the small deviations for largest eigenvalues of the β ensembles and tail inequalities of the general β Tracy-Widom law.     

A note on rates of convergence in the multidimensional CLT for maximum partial sums

In this note we obtain rates of convergence in the central limit theorem for certain maximum of coordinate partial sums of independent identically distributed random vectors having positive mean vector and a nonsingular correlation matrix. The results obtained are in terms of rates of convergence in the multidimensional central limit theorem. Thus under the conditions of Sazonov (1968, Sankhya, Series A30 181–204, Theorem 2), we have the same rate of convergence for the vector of coordinate maximums. Other conditions for the multidimensional CLT are also discussed, c.f., Bhattachaya (1977, Ann. Probability 5 1–27). As an application of one of the results we obtain a multivariate extension of a theorem of Rogozin (1966, Theor. Probability Appl. 11 438–441).     

混合序列部分和的若干收敛性质

本文研究了混合序列部分和的若干收敛性质.利用Serfling不等式推广情形,证明了一类随机变量序列部分和的一个收敛性结果,获得了混合序列部分和的收敛性,并进一步得到了混合序列加权和的强收敛性和完全收敛性,推广并改进了文[2]中有关结果.     

&rho|混合随机变量加权和的收敛性

该文研究了ρ 混合随机变量加权和的强大数律及完全收敛性, 获得了一些新的结果. 该文的结果推广和改进了Bai 等^[1]及Baum 等^[18] 在 i.i.d. 情形时相应的结果, 也推广和改进了Volodin 等^[4]在实值独立时相应的结果. 该文还得到了一关于任意随机变量阵列加权和的完全收敛性定理.     

两两NQD列的Lp收敛性和完全收敛性

在较宽泛的条件下研究了不同分布两两NQD列加权和的收敛性质,利用矩不等式和截尾方法,获得了一般双下标加权系数的加权部分和的LP收敛性和完全收敛性定理,推广了前人的相应结果.     

Complete convergence in mean for double arrays of random variables with values in Banach spaces

The rate of moment convergence of sample sums was investigated by Chow (1988) (in case of real-valued random variables). In 2006, Rosalsky et al. introduced and investigated this concept for case random variable with Banach-valued (called complete convergence in mean of order p). In this paper, we give some new results of complete convergence in mean of order p and its applications to strong laws of large numbers for double arrays of random variables taking values in Banach spaces.     

Empirical processes of multidimensional systems with multiple mixing properties

We establish a multivariate empirical process central limit theorem for stationary R^d-valued stochastic processes (X_i)_i≥1 under very weak conditions concerning the dependence structure of the process. As an application, we can prove the empirical process CLT for ergodic torus automorphisms. Our results also apply to Markov chains and dynamical systems having a spectral gap on some Banach space of functions. Our proof uses a multivariate extension of the techniques introduced by Dehling et al. (2009) [9] in the univariate case. As an important technical ingredient, we prove a 2pth moment bound for partial sums in multiple mixing systems.     

Rates of convergence in the central limit theorem for linear statistics of martingale differences

In this paper, we give rates of convergence for minimal distances between linear statistics of martingale differences and the limiting Gaussian distribution. In particular the results apply to the partial sums of (possibly long range dependent) linear processes, and to the least squares estimator in some parametric regression models.     

Convergence of weighted sums of random elements in D[0, 1]

Convergence in probability for Toeplitz weighted sums is obtained for convex tight random elements in D[0, 1] under pointwise conditions. The almost sure convergence of the weighted sums is proved for independent, convex tight random elements and for independent, identically distributed random elements. Special techniques and concepts are developed in order to obtain these results in the Skorohod topology of D[0, 1].     

L 1 bounds for some martingale central limit theorems

The aim of this paper is to extend the results in [E. Bolthausen, Exact convergence rates in some martingale central limit theorems, Ann. Probab., 10(3):672–688, 1982] and [J.C. Mourrat, On the rate of convergence in the martingale central limit theorem, Bernoulli, 19(2):633–645, 2013] to the L1-distance between distributions of normalized partial sums for martingale-difference sequences and the standard normal distribution.     

COMPLETE CONVERGENCE THEORY OF THE CONTACT PROCESS ON Td × Z1

The author considers the contact process on a branching plane T_d × Z, which is the product of a regular tree T_d and the line Z. It is shown that above the second critical point, the complete convergence theory holds.     

On the Convergence of Partial Sums with Respect to Vilenkin System on the Martingale Hardy Spaces

In this paper, we derive characterizations of boundedness of subsequences of partial sums with respect to Vilenkin system on the martingale Hardy spaces H_p when 0 < p < 1. Moreover, we find necessary and sufficient conditions for the modulus of continuity of martingales f ∈ H_p, which provide convergence of subsequences of partial sums on the martingale Hardy spaces H_p. It is also proved that these results are the best possible in a special sense. As applications, some known and new results are pointed out.     

Complete convergence of weighted sums for arrays of rowwise φ-mixing random variables

In this paper, we establish the complete convergence and complete moment convergence of weighted sums for arrays of rowwise φ-mixing random variables, and the Baum-Katz-type result for arrays of rowwise φ-mixing random variables. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of φ-mixing random variables is obtained. We extend and complement the corresponding results of X. J. Wang, S. H. Hu (2012).     

On complete convergence for weighted sums of asymptotically linear negatively dependent random field

In this paper we establish the complete convergence for weighted sums of asymptotically linear negatively quadrant dependent random field, which contains a linear negatively quadrant dependent field and a ρ^∗${\rho }^{\ast }$-mixing random field.