A NEW METHOD OF NORMAL APPROXIMATION

By Sourav Chatterjee. 2006

195 citations recorded by the source corpus.

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 concentration of measure. Unlike Skorokhod embeddings, the object whose variance must be bounded has an explicit formula that makes it possible to carry out the program more easily. As an application, we derive a general CLT for functions that are obtained as combinations of many local contributions, where the definition of "local" itself depends on the data. Several examples are given, including the solution to a nearest-neighbor CLT problem posed by P. Bickel.

Subfields: Applied Mathematics, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/31273031896cba94aea2553a3a8cb761cdc441c3

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In Applied Mathematics

In Numerical Analysis

In Statistical and Nonlinear Physics

Citations in the corpus

On boosting the power of Chatterjee’s rank correlation (2021, methodology): “(Chatterjee, 2008, Theorem 2.5) Let W be a measurable space and f : Wn → R be a measurable map admitting a symmetric interaction rule G.”
The additive structure of elliptic homogenization (2016, methodology): “In these works, the space is the discrete lattice Zd, d ≥ 3, and a key assumption is that the probability measure has an underlying product structure which makes available tools such as concentration inequalities, the Chatterjee-Stein [9, 10] method of normal approximation and the Helffer-Sjöstrand”
Normal approximation for stabilizing functionals (2017, methodology): “4 of [10] yields a rate of normal convergence for L (q) NG1 (Xn) in the Wasserstein distance dW given by the right-hand side of (3.”
Fluctuations of eigenvalues and second order Poincaré inequalities (2007, methodology): “However, no familiarity with [16] is required here.”
Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization (2014, methodology): “At this stage, it is difficult to compare our techniques with those of [8] for mainly two reasons: (i) while it is possible to apply the results from [8] to compute Berry-Esséen bounds for Poisson functionals (as demonstrated in the recent reference [10], dealing with the total length of the minimal”
From Stein identities to moderate deviations (2009, unknown): “…method to obtain Berry-Esseentype bounds for dependent random variables can be found in, for example, Diaconis (1977), Baldi, Rinott and Stein (1989), Barbour (1990), Dembo and Rinott (1996), Goldstein and Reinert (1997), Chen and Shao (2004), Chatterjee (2008), and Nourdin and Peccati (2009).”
On Stein's Method for Infinitely Divisible Laws with Finite First Moment (2017, background): “several quantitative convergence results follow). (iii) As ﬁrst noticed in [41], the quantity 1 2 Xn k=1 Z+N −N k Kν(t,N) n −K(t,N) dt, is the L1 analog of the L2 Stein discrepancy considered e.g. in [7, 8, 26, 21]. As such, it seems to be the appropriate Stein quantity to deal with sums of independ”
High-dimensional central limit theorems by Stein’s method (2020, methodology): “Next, we consider non-linear statistics along the lines of Chatterjee (2008), Chen and Röllin (2010) and Dung (2019).”
A short survey of Stein's method (2014, methodology): “Indeed, it may be said that the whole thriving area of Stein’s method in Malliavin calculus, pioneered by Nourdin and Peccati [60], is an “abstractification” of the ideas contained in [21] and [22].”
Limit theory for point processes in manifolds (2011, background): “Subject to Var[Rα(n1/mYn)] = (n), [13] yields a rate of convergence in (2.”
Stein's Method and Stochastic Analysis of Rademacher Functionals (2008, unknown): “Technical proofs are deferred to the Appendix.”
Limit theorems for empirical Fr\'echet means of independent and non-identically distributed manifold-valued random variables (2011, background): “…is a substantial literature on central limit theorems and normal approximations for vector-valued random variables; see Bhattacharya and Rao (1976) for an exposition in book form, and more recently Chatterjee (2008) and Röllin (2011) (both of whom describe approaches which apply Stein’s method).”
Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, methodology): “The proofs there are based on “nonlinear” sensitivity estimates such as the Chatterjee-Stein method of normal approximation [31, 32, 103].”
Leave‐Out Estimation of Variance Components (2018, unknown): “The large sample distribution of the estimator is derived using a variant of the arguments in Chatterjee (2008) and Sølvsten (2020).”
MULTIVARIATE NORMAL APPROXIMATION USING EXCHANGEABLE PAIRS (2007, unknown): “Diaconis and Meckes in the survey paper [13], and to a general method of normal approximation for arbitrary functions of independent random variables in [ 12 ].”
Stein’s method and Normal approximation of Poisson functionals (2008, background): “Point (ii) is proved, for example, in [2], Lemma 4.”
On the Kozachenko-Leonenko entropy estimator (2016, background): “The work of Bickel and Breiman has been generalized in many directions, see Chatterjee [5], Penrose and Yukich [19] and Biau and Devroye [3], with new ideas of proofs.”
Stein’s method on Wiener chaos (2007, background): “: exchangeable pairs [49], diffusion generators [3,19], size-bias transforms [20], zero-bias transforms [21], local dependency graphs [10] and graphical-geometric rules [8]).”