Variational Inequalities with Examples and an Application to the Central Limit Theorem

Cacoullos, T.; Papathanasiou, V.; Utev, Sergey

doi:10.1214/aop/1176988616

Cited by 52 publications

(51 citation statements)

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“…This idea already occurs in the literature on normal approximation [15]. However, it is not at all clear how one can infer facts about h(W ) when W is an immensely complex object like a linear statistic of eigenvalues of a Wigner matrix.…”

Section: 2mentioning

confidence: 99%

Fluctuations of eigenvalues and second order Poincaré inequalities

Chatterjee

2007

Probab. Theory Relat. Fields

181

263

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Abstract. Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of 'second order Poincaré inequalities': just as ordinary Poincaré inequalities give variance bounds, second order Poincaré inequalities give central limit theorems. The proof of the main result employs Stein's method of normal approximation. A number of examples are worked out, some of which are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices.

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Section: 2mentioning

confidence: 99%

Fluctuations of eigenvalues and second order Poincaré inequalities

Chatterjee

2007

Probab. Theory Relat. Fields

181

263

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“…More generally, if the collection X α,β satisfies Definition 1.2 for variables indexed over Γ, then the restriction of the same variables indexed over a subset of Γ satisfies Definition 1.2 over that subset. In particular, when Γ is finite we need only verify the definition for I = Γ. Modifying the Stein normal characterization to yield an identity such as (2) which applies to a large class of distributions is an approach also taken in [4]. Rather than changing the distribution of X to satisfy the right hand side of (2), in [4] the existence of a function w is postulated such that EXf (X) = σ 2 Ew(X)f (X).…”

Section: Introductionmentioning

confidence: 99%

“…In particular, when Γ is finite we need only verify the definition for I = Γ. Modifying the Stein normal characterization to yield an identity such as (2) which applies to a large class of distributions is an approach also taken in [4]. Rather than changing the distribution of X to satisfy the right hand side of (2), in [4] the existence of a function w is postulated such that EXf (X) = σ 2 Ew(X)f (X). Based on an idea in [2], the use of the w function is extended in [3] to a multivariate case for independent mean zero variables X 1 , .…”

Section: Introductionmentioning

confidence: 99%

Zero biasing in one and higher dimensions, and applications

Goldstein¹,

Reinert²

2005

Stein's Method and Applications

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Given any mean zero, finite variance σ 2 random variable W , there exists a unique distribution on a variable W * such that EW f (W ) = σ 2 Ef (W * ) for all absolutely continuous functions f for which these expectations exist. This distributional 'zero bias' transformation of W to W * , of which the normal is the unique fixed point, was introduced in [9] to obtain bounds in normal approximations. After providing some background on the zero bias transformation in one dimension, we extend its definition to higher dimension and illustrate with an application to the normal approximation of sums of vectors obtained by simple random sampling.

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“…In the present section we give a simple proof of the following result. It should be noted that several proofs of Prokhorov's theorem (3.1), have appeared in the probability literature the last few years (see for example [1], [13], [6]), but it seems that the authors use some additional restrictive conditions on the density /. The present approach, however, does not require any further assumption, other than a finite second moment.…”

Section: о \F K (T) -(P(t]fjtmentioning

confidence: 96%