Stein's method of exchangeable pairs for the Beta distribution and generalizations

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.

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Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model

The Annals of Applied Probability ◽

10.1214/10-aap712 ◽

2011 ◽

Vol 21 (2) ◽

pp. 464-483 ◽

Cited By ~ 23

Author(s):

Sourav Chatterjee ◽

Qi-Man Shao

Keyword(s):

Stein’S Method ◽

Stein's Method ◽

Exchangeable Pairs

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Stein's Method for the Beta Distribution and the Pólya-Eggenberger Urn

Journal of Applied Probability ◽

10.1017/s0021900200013875 ◽

2013 ◽

Vol 50 (04) ◽

pp. 1187-1205 ◽

Cited By ~ 6

Author(s):

Larry Goldstein ◽

Gesine Reinert

Keyword(s):

Random Walk ◽

Beta Distribution ◽

Simple Random Walk ◽

Return Time ◽

Wasserstein Distance ◽

Stein’S Method ◽

Discrete Distributions ◽

Terminal Point ◽

Optimal Order ◽

Stein's Method

Using a characterizing equation for the beta distribution, Stein's method is applied to obtain bounds of the optimal order for the Wasserstein distance between the distribution of the scaled number of white balls drawn from a Pólya-Eggenberger urn and its limiting beta distribution. The bound is computed by making a direct comparison between characterizing operators of the target and the beta distribution, the former derived by extending Stein's density approach to discrete distributions. In addition, refinements are given to Döbler's (2012) result for the arcsine approximation for the fraction of time a simple random walk of even length spends positive, and so also to the distributions of its last return time to 0 and its first visit to its terminal point, by supplying explicit constants to the present Wasserstein bound and also demonstrating that its rate is of the optimal order.

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Stein's Method for the Beta Distribution and the Pólya-Eggenberger Urn

Journal of Applied Probability ◽

10.1239/jap/1389370107 ◽

2013 ◽

Vol 50 (4) ◽

pp. 1187-1205 ◽

Cited By ~ 2

Author(s):

Larry Goldstein ◽

Gesine Reinert

Keyword(s):

Random Walk ◽

Beta Distribution ◽

Simple Random Walk ◽

Return Time ◽

Wasserstein Distance ◽

Stein’S Method ◽

Discrete Distributions ◽

Terminal Point ◽

Optimal Order ◽

Stein's Method

Using a characterizing equation for the beta distribution, Stein's method is applied to obtain bounds of the optimal order for the Wasserstein distance between the distribution of the scaled number of white balls drawn from a Pólya-Eggenberger urn and its limiting beta distribution. The bound is computed by making a direct comparison between characterizing operators of the target and the beta distribution, the former derived by extending Stein's density approach to discrete distributions. In addition, refinements are given to Döbler's (2012) result for the arcsine approximation for the fraction of time a simple random walk of even length spends positive, and so also to the distributions of its last return time to 0 and its first visit to its terminal point, by supplying explicit constants to the present Wasserstein bound and also demonstrating that its rate is of the optimal order.

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Step Size in Stein's Method of Exchangeable Pairs

Combinatorics Probability Computing ◽

10.1017/s096354830999023x ◽

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.

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