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On the rate of convergence in the central limit theorem for martingale difference sequences  [PDF]
Lahcen Ouchti
Mathematics , 2004, DOI: 10.1016/j.anihpb.2004.03.003
Abstract: We established the rate of convergence in the central limit theorem for stopped sums of a class of martingale difference sequences.
On the central and local limit theorem for martingale difference sequences  [PDF]
Mohamed El Machkouri,Dalibor Volny
Mathematics , 2004,
Abstract: Let $(\Omega, \A, \mu)$ be a Lebesgue space and $T$ an ergodic measure preserving automorphism on $\Omega$ with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on $\Omega$ with a common non-degenerate lattice distribution satisfying the central limit theorem with an arbitrarily slow rate of convergence and not satisfying the local limit theorem. A similar result is established for martingale difference sequences with densities provided the entropy is infinite. In addition, the martingale difference sequence may be chosen to be strongly mixing.
Exact convergence rates in the central limit theorem for a class of martingales  [PDF]
Mohamed El Machkouri,Lahcen Ouchti
Mathematics , 2004,
Abstract: We give optimal convergence rates in the central limit theorem for a large class of martingale difference sequences with bounded third moments. The rates depend on the behaviour of the conditional variances and for stationary sequences the rate $n^{-1/2}\log n$ is reached. We give interesting examples of martingales with unbounded increments which belong to the considered class.
On the rate of convergence and Berry-Esseen type theorems for a multivariate free central limit theorem  [PDF]
Roland Speicher
Mathematics , 2007,
Abstract: We address the question of a Berry Esseen type theorem for the speed of convergence in a multivariate free central limit theorem. For this, we estimate the difference between the operator-valued Cauchy transforms of the normalized partial sums in an operator-valued free central limit theorem and the Cauchy transform of the limiting operator-valued semicircular element.
A uniform estimate for the rate of convergence in the multidimensional central limit theorem for homogeneous Markov chains  [cached]
M. Gharib
International Journal of Mathematics and Mathematical Sciences , 1996, DOI: 10.1155/s0161171296000634
Abstract: In this paper a uniform estimate is obtained for the remainder term in the central limit theorem (CLT) for a sequence of random vectors forming a homogeneous Markov chain with arbitrary set of states. The result makes it possible to estimate the rate of convergence in the CLT without assuming the finiteness of the absolute third moment of the transition probabilities. Some consequences are also proved.
Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen gas and to the Sinai billiard  [PDF]
Fran?oise Pène
Mathematics , 2006, DOI: 10.1214/105051605000000476
Abstract: We show how Rio's method [Probab. Theory Related Fields 104 (1996) 255--282] can be adapted to establish a rate of convergence in ${\frac{1}{\sqrt{n}}}$ in the multidimensional central limit theorem for some stationary processes in the sense of the Kantorovich metric. We give two applications of this general result: in the case of the Knudsen gas and in the case of the Sinai billiard.
Uniform Convergence and the Central Limit Theorem  [PDF]
John D. Williams
Mathematics , 2011, DOI: 10.1007/s11785-010-0100-7
Abstract: We prove results about uniform convergence of densities in the free central limit theorem without assumptions of boundedness on the support.
On martingale tail sums for the path length in random trees  [PDF]
Henning Sulzbach
Computer Science , 2014,
Abstract: For a martingale $(X_n)$ converging almost surely to a random variable $X$, the sequence $(X_n - X)$ is called martingale tail sum. Recently, Neininger [Random Structures Algorithms, 46 (2015), 346-361] proved a central limit theorem for the martingale tail sum of R{\'e}gnier's martingale for the path length in random binary search trees. Gr{\"u}bel and Kabluchko [2014, preprint, arXiv 1410.0469] gave an alternative proof also conjecturing a corresponding law of the iterated logarithm. We prove the central limit theorem with convergence of higher moments and the law of the iterated logarithm for a family of trees containing binary search trees, recursive trees and plane-oriented recursive trees.
Exact convergence rate and leading term in central limit theorem for student's t statistic  [PDF]
Peter Hall,Qiying Wang
Mathematics , 2004, DOI: 10.1214/009117904000000252
Abstract: The leading term in the normal approximation to the distribution of Student's t statistic is derived in a general setting, with the sole assumption being that the sampled distribution is in the domain of attraction of a normal law. The form of the leading term is shown to have its origin in the way in which extreme data influence properties of the Studentized sum. The leading-term approximation is used to give the exact rate of convergence in the central limit theorem up to order n^{-1/2}, where n denotes sample size. It is proved that the exact rate uniformly on the whole real line is identical to the exact rate on sets of just three points. Moreover, the exact rate is identical to that for the non-Studentized sum when the latter is normalized for scale using a truncated form of variance, but when the corresponding truncated centering constant is omitted. Examples of characterizations of convergence rates are also given. It is shown that, in some instances, their validity uniformly on the whole real line is equivalent to their validity on just two symmetric points.
Conditional central limit theorem via martingale approximation  [PDF]
Magda Peligrad
Mathematics , 2010,
Abstract: In this paper we survey and further study partial sums of a stationary process via approximation with a martingale with stationary differences. Such an approximation is useful for transferring from the martingale to the original process the conditional central limit theorem. We study both approximations in L_2 and in L_1. The results complement the work of Dedecker Merlevede and Volny (2007), Zhao and Woodroofe (2008), Gordin and Peligrad (2009). The method provides an unitary treatment of many limiting results for dependent random variables including classes of mixing sequences, additive functionals of Markov chains and linear processes.
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