MULTIVARIATE NORMAL APPROXIMATION USING EXCHANGEABLE PAIRS

By Sourav Chatterjee, Elizabeth Meckes. 2007

205 citations recorded by the source corpus.

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 versions of a theorem for situations involving continuous symmetry groups. Our main applications are proofs of the approximate normality of rank k projections of Haar measure on the orthogonal and unitary groups, when k = o(n).

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e949b35d5d08bb2a9f61ec94560f4558701f2660

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Statistics and Probability

Citations in the corpus

On Stein's method for multivariate normal approximation (2009, methodology): “The approach taken in the published version of [2] is to give smoothness conditions instead by requiring bounds on the quantities Mr(h) := sup x∈Rk ‖Dh(x)‖op,”
High-dimensional central limit theorems by Stein’s method (2020, methodology): “First, we consider the exchangeable pair approach developed in Stein (1986) in one-dimensional normal approximations and Chatterjee and Meckes (2008) and Reinert and Röllin (2009) for multivariate normal approximations.”
The Random Matrix Theory of the Classical Compact Groups (2019, methodology): “22 gives a formula for computing mixed moments up to order four of entries of a random orthogonal matrix by exploiting the symmetries of Haar measure; the analog for the unitary group can be found in [18].”
Nearly optimal central limit theorem and bootstrap approximations in high dimensions (2020, methodology): “5), we employ the exchangeable pair approach in Stein’s method for multivariate normal approximation by [11] and [36] along with a symmetry argument by [23, 24] (cf.”
Convergence in total variation on Wiener chaos (2012, unknown): “In a non-discrete framework (which is the case here), it is fairly understood that this method can give good results with respect to the total variation distance only in dimension one (see [4]) and when the target law is Gaussian (see [5]).”
Normal Approximation by Stein's Method (2010, background): “• BR76 = Bhattacharya and Ranga Rao (1976) • RR96 = Rinott and Rotar (1996) and RR97= Rinott and Rotar (1997) • MT = Meckes (2006) Ph.D. thesis • Alea = Chatterjee and Meckes (2008) • Luminy = Meckes (2009), Luminy paper • PTTM = Pollard (2029) and UGMTP = Pollard (2001).”
Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors (2012, background): “[8]), p is often comparable or even larger than n, and the classical CLT does not apply in such cases.”
Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition (2007, methodology): “Compared to the main theorem of Chatterjee and Meckes (2008), which only needs the existence of two derivatives, our Theorem 2.1 is more restrictive in the choice of test functions h.”
A short survey of Stein's method (2014, methodology): “In a joint paper [28] with Meckes in 2008, we found a way to implement the method of exchangeable pairs in the multivariate setting.”
Estimating the unseen: an n/log(n)-sample estimator for entropy and support size, shown optimal via new CLTs (2011, methodology): “In particular, there have been several successful applications of Stein’s method in multivariate settings [17, 22, 36].”
Multivariate normal approximation using Stein's method and Malliavin calculus (2008, methodology): “We can now use the same arguments as in the proof of Lemma 3 in [4] to deduce that”
Gaussian approximation of suprema of empirical processes (2012, methodology): “The following argument is inspired by [7], Theorem 7.”
Measuring Sample Quality with Diffusions (2016, background): “11 Example 1 (Bayesian logistic regression with Gaussian prior).”
Multi-Dimensional Gaussian Fluctuations on the Poisson Space (2010, methodology): “The following result is a d-dimensional version of Stein’s Lemma; analogous statements can be found in [4, 10, 22] – see also Barbour [1] and Götze [5], in connection with the so-called “generator approach” to Stein’s method.”