On the dependence of the Berry–Esseen bound on dimension

Bentkus, V.

doi:10.1016/s0378-3758(02)00094-0

Cited by 169 publications

(222 citation statements)

References 5 publications

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“…Inequality (3.4) provides a Berry-Esseen type bound which depends on α, on the second moment of the density f but not on the dimension d. This is in contrast to a series of known results where the right hand side of (3.4) depends on d. More precisely, Bentkus (2003) where C is the class of convex sets. The bound in (3.5) contains the best known dependence on the dimension under those assumptions.…”

Section: The Finite Dimensional Case: a Dimension Independent Berry-ementioning

confidence: 92%

A note on a local limit theorem for Wiener space valued random variables

Lanconelli¹,

Stan²

2016

Bernoulli

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We prove a local limit theorem, i.e. a central limit theorem for densities, for a sequence of independent and identically distributed random variables taking values on an abstract Wiener space; the common law of those random variables is assumed to be absolutely continuous with respect to the reference Gaussian measure. We begin by showing that the key roles of scaling operator and convolution product in this infinite dimensional Gaussian framework are played by the Ornstein-Uhlenbeck semigroup and Wick product, respectively. We proceed by establishing a necessary condition on the density of the random variables for the local limit theorem to hold true. We then reverse the implication and prove under an additional assumption the desired L 1 -convergence of the density of X 1 +···+Xn √ n . We close the paper comparing our result with certain Berry-Esseen bounds for multidimensional central limit theorems.

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Section: The Finite Dimensional Case: a Dimension Independent Berry-ementioning

confidence: 92%

A note on a local limit theorem for Wiener space valued random variables

Lanconelli¹,

Stan²

2016

Bernoulli

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“…(ii) They allow for arbitrary covariance structures between the coordinates in Gaussian random vectors, and (iii) they are sharp in the sense that there is an example for which the bound is tight up to a dimension independent constant. We note that these anti-concentration bounds are sharper than those that result from application of the universal reverse isoperimetric inequality of [2] (see also [3], p.386-367).…”

Section: Introductionmentioning

confidence: 75%

Comparison and anti-concentration bounds for maxima of Gaussian random vectors

Chernozhukov

Chetverikov

Kato

2016

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Abstract. Slepian and Sudakov-Fernique type inequalities, which compare expectations of maxima of Gaussian random vectors under certain restrictions on the covariance matrices, play an important role in probability theory, especially in empirical process and extreme value theories. Here we give explicit comparisons of expectations of smooth functions and distribution functions of maxima of Gaussian random vectors without any restriction on the covariance matrices. We also establish an anticoncentration inequality for the maximum of a Gaussian random vector, which derives a useful upper bound on the Lévy concentration function for the Gaussian maximum. The bound is dimension-free and applies to vectors with arbitrary covariance matrices. This anti-concentration inequality plays a crucial role in establishing bounds on the Kolmogorov distance between maxima of Gaussian random vectors. These results have immediate applications in mathematical statistics. As an example of application, we establish a conditional multiplier central limit theorem for maxima of sums of independent random vectors where the dimension of the vectors is possibly much larger than the sample size.

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“…Essentially, we are interested in characterizing the (n, ε)-optimal rate region for the WAK problem, the (n, ε)-Wyner-Ziv rate-distortion function and the (n, ε)-capacity of GP problem up to the second-order term. We do this by applying the multidimensional Berry-Esséen theorem [21], [50] to the finite blocklength CS-type bounds in Corollaries 6, 9 and 11. Throughout, we will not concern ourselves with optimizing the third-order terms.…”

Section: Achievable Second-order Coding Ratesmentioning

confidence: 99%

Nonasymptotic and Second-Order Achievability Bounds for Coding With Side-Information

Watanabe

Kuzuoka

Tan

2015

IEEE Trans. Inform. Theory

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Abstract-We present novel non-asymptotic or finite blocklength achievability bounds for three side-information problems in network information theory. These include (i) the WynerAhlswede-Körner (WAK) problem of almost-lossless source coding with rate-limited side-information, (ii) the Wyner-Ziv (WZ) problem of lossy source coding with side-information at the decoder and (iii) the Gel'fand-Pinsker (GP) problem of channel coding with noncausal state information available at the encoder. The bounds are proved using ideas from channel simulation and channel resolvability. Our bounds for all three problems improve on all previous non-asymptotic bounds on the error probability of the WAK, WZ and GP problems-in particular those derived by Verdú. Using our novel non-asymptotic bounds, we recover the general formulas for the optimal rates of these side-information problems. Finally, we also present achievable second-order coding rates by applying the multidimensional Berry-Esséen theorem to our new non-asymptotic bounds. Numerical results show that the second-order coding rates obtained using our non-asymptotic achievability bounds are superior to those obtained using existing finite blocklength bounds.

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On the dependence of the Berry–Esseen bound on dimension

Cited by 169 publications

References 5 publications

A note on a local limit theorem for Wiener space valued random variables

A note on a local limit theorem for Wiener space valued random variables

Comparison and anti-concentration bounds for maxima of Gaussian random vectors

Nonasymptotic and Second-Order Achievability Bounds for Coding With Side-Information

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