Fisher information and the central limit theorem

Bobkov, Sergey G.; Chistyakov, G. P.; Götze, Friedrich

doi:10.1007/s00440-013-0500-5

Cited by 36 publications

(43 citation statements)

References 20 publications

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“…Barron [2] studied the convergence in relative entropy, whereas Shimizu [17] and Johnson and Barron [10] studied the convergence in Fisher information. As far as rates of convergence are concerned, one can quote Mamatov and Sirazdinov [13] for the total variation distance and, very recently, Bobkov, Chistyakov and Götze for bounds in entropy [3], in Fisher information [4] and for Edgeworth-type expansions in the entropic central limit theorem [5]. Finally we mention [6,7] for a variational approach of these issues with some variance bounds.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

An invariance principle under the total variation distance

Nourdin

Poly

2015

Stochastic Processes and their Applications

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Let X 1 , X 2 , . . . be a sequence of i.i.d. random variables, with mean zero and variance one. Let W n = (X 1 + . . . + X n )/ √ n. An old and celebrated result of Prohorov [16] asserts that W n converges in total variation to the standard Gaussian distribution if and only if W n 0 has an absolutely continuous component for some n 0 . In the present paper, we give yet another proof and extend Prohorov's theorem to a situation where, instead of W n , we consider more generally a sequence of homogoneous polynomials in the X i . More precisely, we exhibit conditions for a recent invariance principle proved by Mossel, O'Donnel and Oleszkiewicz [14] to hold under the total variation distance. There are many works about CLT under various metrics in the literature, but the present one seems to be the first attempt to deal with homogeneous polynomials in the X i with degree strictly greater than one.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

An invariance principle under the total variation distance

Nourdin

Poly

2015

Stochastic Processes and their Applications

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“…Convergence of densities in uniform distance (also in total variation) has also been studied using Fisher information theory via Shimizu's inequality (see for instance, [29,2,3,8])…”

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confidence: 99%

Convergence of densities of some functionals of Gaussian processes

Nualart

2014

Journal of Functional Analysis

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The aim of this paper is to establish the uniform convergence of the densities of a sequence of random variables, which are functionals of an underlying Gaussian process, to a normal density. Precise estimates for the uniform distance are derived by using the techniques of Malliavin calculus, combined with Stein's method for normal approximation. We need to assume some non-degeneracy conditions. First, the study is focused on random variables in a fixed Wiener chaos, and later, the results are extended to the uniform convergence of the derivatives of the densities and to the case of random vectors in some fixed chaos, which are uniformly non-degenerate in the sense of Malliavin calculus. Explicit upper bounds for the uniform norm are obtained for random variables in the second Wiener chaos, and an application to the convergence of densities of the least square estimator for the drift parameter in Ornstein-Uhlenbeck processes is discussed.

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“…Starting from the pioneering work of Linnik [34], the role of Fisher information to obtain alternative proofs of the central limit theorem has been enlightened by a number of papers (cf. [1,2,3,9,10,13,28,29,36,44] and the references therein). Only recently, Bobkov, Chistyakov and Götze [10] used of the relative Fisher information to study convergence to stable laws.…”

Section: Discussionmentioning

confidence: 99%

The fractional Fisher information and the central limit theorem for stable laws

Toscani

2015

Ricerche mat.

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Abstract. A new information-theoretic approach to the central limit theorem for stable laws is presented. The main novelty is the concept of relative fractional Fisher information, which shares most of the properties of the classical one, included Blachman-Stam type inequalities. These inequalities relate the fractional Fisher information of the sum of n independent random variables to the information contained in sums over subsets containing n − 1 of the random variables. As a consequence, a simple proof of the monotonicity of the relative fractional Fisher information in central limit theorems for stable law is obtained, together with an explicit decay rate.

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Fisher information and the central limit theorem

Cited by 36 publications

References 20 publications

An invariance principle under the total variation distance

An invariance principle under the total variation distance

Convergence of densities of some functionals of Gaussian processes

The fractional Fisher information and the central limit theorem for stable laws

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