• Corpus ID: 8815805

MULTIVARIATE NORMAL APPROXIMATION USING EXCHANGEABLE PAIRS

@article{Chatterjee2007MULTIVARIATENA,
  title={MULTIVARIATE NORMAL APPROXIMATION USING EXCHANGEABLE PAIRS},
  author={Sourav Chatterjee and Elizabeth S. Meckes},
  journal={arXiv: Probability},
  year={2007}
}
Since the introduction of Stein's method in the early 1970s, much research has been done in extending and strengthening it; however, there does not exist a version of Stein's original method of exchangeable pairs for multivariate normal approximation. The aim of this article is to fill this void. We present three abstract normal approximation theorems using exchangeable pairs in multivariate contexts, one for situations in which the underlying symmetries are discrete, and real and complex… 
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References

SHOWING 1-10 OF 63 REFERENCES
A NEW METHOD OF NORMAL APPROXIMATION
We introduce a new version of Stein's method that reduces a large class of normal approximation problems to variance bounding exercises, thus making a connection between central limit theorems and
Stein's Method: Expository Lectures and Applications
TLDR
A review of Stein’s method applied to the case of discrete random variables and attempt to complete one of Stein's open problems, that of providing a discrete version for chapter 6 of his book.
Poisson Approximation and the Chen-Stein Method
The Chen-Stein method of Poisson approximation is a powerful tool for computing an error bound when approximating probabilities using the Poisson distribution. In many cases, this bound may be given
Stein’s method for birth and death chains
This article presents a review of Stein's method applied to the case of discrete random variables. We attempt to complete one of Stein's open problems, that of providing a discrete version for
Stein’s method for Markov chains: first examples
Charles Stein has introduced a general approach to proving approx- imation theorems in probability theory. The method is being actively used for normal and Poisson approximation. This paper uses the
Two moments su ce for Poisson approx-imations: the Chen-Stein method
Convergence to the Poisson distribution, for the number of occurrences of dependent events, can often be established by computing only first and second moments, but not higher ones. This remarkable
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 multidimensional
Exchangeable pairs and Poisson approximation
This is a survery paper on Poisson approximation using Stein's method of exchangeable pairs. We illustrate using Poisson-binomial trials and many variations on three classical problems of
Finite de Finetti theorems in linear models and multivariate analysis
Let Xl,-.. , Xk be a sequence of random vectors. We give symmetry conditions on the joint distribution which imply that it is well approximated by a mixture of normal distributions. Examples include
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