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.

scholarly journals Step Size in Stein's Method of Exchangeable Pairs

2009 ◽  
Vol 18 (6) ◽  
pp. 979-1017
Author(s):  
NATHAN ROSS
Keyword(s):  
Normal Distribution ◽  
Convergence Rates ◽  
Normal Approximation ◽  
Poisson Approximation ◽  
Stein’S Method ◽  
Stein's Method ◽  
Step Size ◽  
Rigorous Results ◽  
Exchangeable Pairs ◽  
Natural Way

Stein's method of exchangeable pairs is examined through five examples in relation to Poisson and normal distribution approximation. In particular, in the case where the exchangeable pair is constructed from a reversible Markov chain, we analyse how modifying the step size of the chain in a natural way affects the error term in the approximation acquired through Stein's method. It has been noted for the normal approximation that smaller step sizes may yield better bounds, and we obtain the first rigorous results that verify this intuition. For the examples associated to the normal distribution, the bound on the error is expressed in terms of the spectrum of the underlying chain, a characteristic of the chain related to convergence rates. The Poisson approximation using exchangeable pairs is less studied than the normal, but in the examples presented here the same principles hold.

Stein's method and poisson process convergence

1988 ◽  
Vol 25 (A) ◽  
pp. 175-184 ◽  
Author(s):  
A. D. Barbour
Keyword(s):  
Point Processes ◽  
Rates Of Convergence ◽  
Poisson Approximation ◽  
Stein’S Method ◽  
Simple Problem ◽  
Stein's Method ◽  
General Technique ◽  
One Dimensional ◽  
Process Convergence ◽  
Process Approximation

Stein's method of obtaining rates of convergence, well known in normal and Poisson approximation, is considered here in the context of approximation by Poisson point processes, rather than their one-dimensional distributions. A general technique is sketched, whereby the basic ingredients necessary for the application of Stein's method may be derived, and this is applied to a simple problem in Poisson point process approximation.

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

scholarly journals Solving the Stein Equation in compound poisson approximation

1998 ◽  
Vol 30 (02) ◽  
pp. 449-475 ◽  
Author(s):  
A. D. Barbour ◽  
Sergey Utev
Keyword(s):  
Poisson Approximation ◽  
Stein’S Method ◽  
Restricted Range ◽  
Stein's Method ◽  
Kolmogorov Distance ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Stein Equation ◽  
The Mean ◽  
Accuracy Of Approximation

The accuracy of compound Poisson approximation can be estimated using Stein's method in terms of quantities similar to those which must be calculated for Poisson approximation. However, the solutions of the relevant Stein equation may, in general, grow exponentially fast with the mean number of ‘clumps’, leading to many applications in which the bounds are of little use. In this paper, we introduce a method for circumventing this difficulty. We establish good bounds for those solutions of the Stein equation which are needed to measure the accuracy of approximation with respect to Kolmogorov distance, but only in a restricted range of the argument. The restriction on the range is then compensated by a truncation argument. Examples are given to show that the method clearly outperforms its competitors, as soon as the mean number of clumps is even moderately large.

Sign in / Sign up

Export Citation Format

Share Document