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.

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.

On the rate of Poisson process approximation to a Bernoulli process

1997 ◽  
Vol 34 (4) ◽  
pp. 898-907 ◽  
Author(s):  
Aihua Xia
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Discrete Time ◽  
Wasserstein Distance ◽  
Stein’S Method ◽  
Upper Bounds ◽  
Stein's Method ◽  
Bernoulli Process ◽  
Logarithmic Factor ◽  
Process Approximation

This note gives the rate for a Wasserstein distance between the distribution of a Bernoulli process on discrete time and that of a Poisson process, using Stein's method and Palm theory. The result here highlights the possibility that the logarithmic factor involved in the upper bounds established by Barbour and Brown (1992) and Barbour et al. (1995) may be superfluous in the true Wasserstein distance between the distributions of a point process and a Poisson process.

scholarly journals Power Laws in Preferential Attachment Graphs and Stein's Method for the Negative Binomial Distribution

2013 ◽  
Vol 45 (03) ◽  
pp. 876-893 ◽  
Author(s):  
Nathan Ross
Keyword(s):  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Preferential Attachment ◽  
Power Laws ◽  
Rates Of Convergence ◽  
Stein’S Method ◽  
Stein's Method ◽  
Power Law Distribution ◽  
New Formulation

For a family of linear preferential attachment graphs, we provide rates of convergence for the total variation distance between the degree of a randomly chosen vertex and an appropriate power law distribution as the number of vertices tends to ∞. Our proof uses a new formulation of Stein's method for the negative binomial distribution, which stems from a distributional transformation that has the negative binomial distributions as the only fixed points.

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.

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