scholarly journals Stein’s method and Normal approximation of Poisson functionals

2010 ◽  
Vol 38 (2) ◽  
pp. 443-478 ◽  
Author(s):  
G. Peccati ◽  
J. L. Solé ◽  
M. S. Taqqu ◽  
F. Utzet
Keyword(s):  
Normal Approximation ◽  
Stein’S Method ◽  
Stein's Method

scholarly journals Multivariate normal approximations by Stein's method and size bias couplings

1996 ◽  
Vol 33 (01) ◽  
pp. 1-17 ◽  
Author(s):  
Larry Goldstein ◽  
Yosef Rinott
Keyword(s):  
Normal Approximation ◽  
Stein’S Method ◽  
Dependence Structure ◽  
Stein's Method ◽  
Multivariate Normal ◽  
Nonlinear Functions ◽  
Normal Approximations ◽  
Multivariate Normal Approximation ◽  
Non Local ◽  
Size Bias

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. Theorem 1.2 requires multivariate size bias coupling, which we discuss in studying the approximation of distributions of sums of dependent random vectors. In the univariate case, we briefly illustrate this approach for certain sums of nonlinear functions of multivariate normal variables. As a second illustration, we show that the multivariate distribution counting the number of vertices with given degrees in certain random graphs is asymptotically multivariate normal and obtain a bound on the rate of convergence. Both examples demonstrate that this approach may be suitable for situations involving non-local dependence. We also present Theorem 1.4 for sums of vectors having a local type of dependence. We apply this theorem to obtain a multivariate normal approximation for the distribution of the random p-vector, which counts the number of edges in a fixed graph both of whose vertices have the same given color when each vertex is colored by one of p colors independently. All normal approximation results presented here do not require an ordering of the summands related to the dependence structure. This is in contrast to hypotheses of classical central limit theorems and examples, which involve for example, martingale, Markov chain or various mixing assumptions.

scholarly journals A Nonuniform Bound to an Independent Test in High Dimensional Data Analysis via Stein’s Method

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Nahathai Rerkruthairat
Keyword(s):  
Normal Approximation ◽  
Correlation Coefficients ◽  
Random Variable ◽  
Stein’S Method ◽  
High Dimensional ◽  
Stein's Method ◽  
Real Numbers ◽  
Complete Independence ◽  
Independent Test ◽  
Sample Correlation

The Berry-Esseen bound for the random variable based on the sum of squared sample correlation coefficients and used to test the complete independence in high diemensions is shown by Stein’s method. Although the Berry-Esseen bound can be applied to all real numbers in R, a nonuniform bound at a real number z usually provides a sharper bound if z is fixed. In this paper, we present the first version of a nonuniform bound on a normal approximation for this random variable with an optimal rate of 1/0.5+|z|·O1/m by using Stein’s method.

scholarly journals Discretized normal approximation by Stein’s method

Bernoulli ◽  
2014 ◽  
Vol 20 (3) ◽  
pp. 1404-1431 ◽  
Author(s):  
Xiao Fang
Keyword(s):  
Normal Approximation ◽  
Stein’S Method ◽  
Stein's Method

Two new proofs of the Erdös–Kac Theorem, with bound on the rate of convergence, by Stein's method for distributional approximations

2009 ◽  
Vol 147 (1) ◽  
pp. 95-114 ◽  
Author(s):  
ADAM J. HARPER
Keyword(s):  
Rate Of Convergence ◽  
Normal Approximation ◽  
Poisson Approximation ◽  
Stein’S Method ◽  
Stein's Method ◽  
Quantitative Form ◽  
Approximation Argument ◽  
Exchangeable Pairs ◽  
Distributional Approximations

AbstractIn this paper, we apply Stein's method for distributional approximations to prove a quantitative form of the Erdös–Kac Theorem. We obtain our best bound on the rate of convergence, on the order of log log log n (log log n)−1/2, by making an intermediate Poisson approximation; we believe that this approach is simpler and more probabilistic than others, and we also obtain an explicit numerical value for the constant implicit in the bound. Different ways of applying Stein's method to prove the Erdös–Kac Theorem are discussed, including a Normal approximation argument via exchangeable pairs, where the suitability of a Poisson approximation naturally suggests itself.

Normal Approximation by Stein’s Method

2011 ◽  
Author(s):  
Louis H.Y. Chen ◽  
Larry Goldstein ◽  
Qi-Man Shao
Keyword(s):  
Normal Approximation ◽  
Stein’S Method ◽  
Stein's Method

Stein’s method of normal approximation for dynamical systems

2019 ◽  
Vol 20 (04) ◽  
pp. 2050021 ◽  
Author(s):  
Olli Hella ◽  
Juho Leppänen ◽  
Mikko Stenlund
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Dynamical Systems ◽  
Central Limit ◽  
Normal Approximation ◽  
Stein’S Method ◽  
Stein's Method ◽  
Multiplicative Constant ◽  
Correlation Decay ◽  
The Central Limit Theorem

We present an adaptation of Stein’s method of normal approximation to the study of both discrete- and continuous-time dynamical systems. We obtain new correlation-decay conditions on dynamical systems for a multivariate central limit theorem augmented by a rate of convergence. We then present a scheme for checking these conditions in actual examples. The principal contribution of our paper is the method, which yields a convergence rate essentially with the same amount of work as the central limit theorem, together with a multiplicative constant that can be computed directly from the assumptions.

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