On normal approximation rates for certain sums of dependent random variables

  title={On normal approximation rates for certain sums of dependent random variables},
  author={Yosef Rinott},
  journal={Journal of Computational and Applied Mathematics},
  • Y. Rinott
  • Published 21 November 1994
  • Mathematics
  • Journal of Computational and Applied Mathematics
Abstract Let X1, …, Xn be dependent random variables, and set λ = E∑ni=1Xi, and σ2 = Var∑ni=1Xi. In most of the applications of Stein's method for normal approximations, the error rate |P((∑ni=1Xi − λ)/σ ⩽ w) − Φ(w)| is of the order of σ− 1 2 . This rate was improved by Stein (1986) and others in some special cases. In this paper it is shown that for certain bounded random variables, a simple refinement of error-term calculations in Stein's method leads to improved rates. 
Multivariate normal approximations by Stein's method and size bias couplings
Stein's method is used to obtain two theorems on multivariate normal approximation. Our main theorem, Theorem 1.2, provides a bound on the distance to normality for any non-negative random vector.Expand
A Multivariate CLT for Decomposable Random Vectors with Finite Second Moments
Stein's method is used to derive a CLT for dependent random vectors possessing the dependence structure from Barbour et al. J. Combin. Theory Ser. B47, 125–145, but under the assumption of secondExpand
Stein's method for normal approximation
Stein’s method originated in 1972 in a paper in the Proceedings of the Sixth Berkeley Symposium. In that paper, he introduced the method in order to determine the accuracy of the normal approximationExpand
Mod-ϕ Convergence, II: Estimates on the Speed of Convergence
In this paper, we give estimates for the speed of convergence towards a limiting stable law in the recently introduced setting of mod-ϕ convergence. Namely, we define a notion of zone of control,Expand
Normal Approximation by Stein ’ s Method
The aim of this paper is to give an overview of Stein’s method, which has turned out to be a powerful tool for estimating the error in normal, Poisson and other approximations, especially for sums ofExpand
Quantitative normal approximation for sums of random variables with multilevel local dependence structure
We establish a quantitative normal approximation result for sums of random variables with multilevel local dependencies. As a corollary, we obtain a quantitative normal approximation result forExpand
A Multivariate CLT for Local Dependence withn -1/2 log nRate and Applications to Multivariate Graph Related Statistics
This paper concerns the rate of convergence in the central limit theorem for certain local dependence structures. The main goal of the paper is to obtain estimates of the rate in the multidimensionalExpand
Stein's method and the zero bias transformation with application to simple random sampling
Let W be a random variable with mean zero and variance 2 . The distribution of a variate W , satisfying EWf(W) = 2 Ef 0 (W ) for smooth functions f, exists uniquely and defines the zero biasExpand
On Edgeworth expansions for dependency-neighborhoods chain structures and Stein's method
Let W be the sum of dependent random variables, and h(x) be a function. This paper provides an Edgeworth expansion of an arbitrary ``length'' for %E{h(W)} in terms of certain characteristics ofExpand
Normal Approximation by Stein's Method
Preface.- 1.Introduction.- 2.Fundamentals of Stein's Method.- 3.Berry-Esseen Bounds for Independent Random Variables.- 4.L^1 Bounds.- 5.L^1 by Bounded Couplings.- 6 L^1: Applications.- 7.Non-uniformExpand


A central limit theorem for decomposable random variables with applications to random graphs
Stein's method of obtaining rates of convergence to the normal distribution is illustrated in the context of random graph theory and results are obtained for the number of copies of a given graph G in K. Expand
A Normal Approximation for the Number of Local Maxima of a Random Function on a Graph
Publisher Summary This chapter discusses the normal approximation for the number of local maxima of a random function on a graph. It discusses the conditions for the approximate normality of theExpand
On Normal Approximations of Distributions in Terms of Dependency Graphs
L'auteur interprete les bornes de l'erreur, introduites par Stein, dans l'approximation normale de sommes de variables aleatoires dependantes en termes de graphes de dependance. Ceci mene a desExpand
Approximate computation of expectations