scholarly journals Rate of convergence to the Circular Law via smoothing inequalities for log-potentials

2020 ◽  
pp. 2150026
Author(s):  
Friedrich Götze ◽  
Jonas Jalowy
Keyword(s):  
Rate Of Convergence ◽  
Optimal Rate ◽  
Empirical Measure ◽  
Random Polynomials ◽  
Kolmogorov Distance ◽  
Empirical Spectral Distribution ◽  
Logarithmic Potentials ◽  
Optimal Rate Of Convergence ◽  
Circular Law ◽  
Ginibre Ensemble

The aim of this paper is to investigate the Kolmogorov distance of the Circular Law to the empirical spectral distribution of non-Hermitian random matrices with independent entries. The optimal rate of convergence is determined by the Ginibre ensemble and is given by [Formula: see text]. A smoothing inequality for complex measures that quantitatively relates the uniform Kolmogorov-like distance to the concentration of logarithmic potentials is shown. Combining it with results from Local Circular Laws, we apply it to prove nearly optimal rate of convergence to the Circular Law in Kolmogorov distance. Furthermore, we show that the same rate of convergence holds for the empirical measure of the roots of Weyl random polynomials.

scholarly journals Empirical measures: regularity is a counter-curse to dimensionality

2020 ◽  
Vol 24 ◽  
pp. 408-434
Author(s):  
Benoît R. Kloeckner
Keyword(s):  
Markov Chains ◽  
Rate Of Convergence ◽  
Decomposition Method ◽  
Classical Method ◽  
Convergence Speed ◽  
Optimal Rate ◽  
Empirical Measure ◽  
Empirical Measures ◽  
Optimal Rate Of Convergence ◽  
Speed Up

We propose a “decomposition method” to prove non-asymptotic bound for the convergence of empirical measures in various dual norms. The main point is to show that if one measures convergence in duality with sufficiently regular observables, the convergence is much faster than for, say, merely Lipschitz observables. Actually, assuming s derivatives with s > d∕2 (d the dimension) ensures an optimal rate of convergence of 1/√n (n the number of samples). The method is flexible enough to apply to Markov chains which satisfy a geometric contraction hypothesis, assuming neither stationarity nor reversibility, with the same convergence speed up to a power of logarithm factor. Our results are stated as controls of the expected distance between the empirical measure and its limit, but we explain briefly how the classical method of bounded difference can be used to deduce concentration estimates.

scholarly journals Davis-type theorems for martingale difference sequences

2005 ◽  
Vol 2005 (2) ◽  
pp. 159-165 ◽  
Author(s):  
George Stoica
Keyword(s):  
Rate Of Convergence ◽  
Classical Case ◽  
Moderate Deviation ◽  
Sharp Contrast ◽  
Optimal Rate ◽  
Martingale Difference ◽  
Normalization Factor ◽  
Second Moment ◽  
Optimal Rate Of Convergence ◽  
Difference Sequences

We study Davis-type theorems on the optimal rate of convergence of moderate deviation probabilities. In the case of martingale difference sequences, under the finite pth moments hypothesis (1≤p<∞), and depending on the normalization factor, our results show that Davis' theorems either hold if and only if p>2 or fail for all p≥1. This is in sharp contrast with the classical case of i.i.d. centered sequences, where both Davis' theorems hold under the finite second moment hypothesis (or less).

scholarly journals ESTIMATING THE QUADRATIC VARIATION SPECTRUM OF NOISY ASSET PRICES USING GENERALIZED FLAT-TOP REALIZED KERNELS

2016 ◽  
Vol 33 (6) ◽  
pp. 1457-1501 ◽  
Author(s):  
Rasmus Tangsgaard Varneskov
Keyword(s):  
Rate Of Convergence ◽  
Asymptotic Variance ◽  
Asymptotic Properties ◽  
Bias Reduction ◽  
Quadratic Variation ◽  
Optimal Rate ◽  
Good Efficiency ◽  
Optimal Rate Of Convergence ◽  
Integrated Variance ◽  
Exogenous Noise

This paper analyzes a generalized class of flat-top realized kernel estimators for the quadratic variation spectrum, that is, the decomposition of quadratic variation into integrated variance and jump variation. The underlying log-price process is contaminated by additive noise, which consists of two orthogonal components to accommodate α-mixing dependent exogenous noise and an asymptotically non-degenerate endogenous correlation structure. In the absence of jumps, the class of estimators is shown to be consistent, asymptotically unbiased, and mixed Gaussian at the optimal rate of convergence, n1/4. Exact bounds on lower-order terms are obtained, and these are used to propose a selection rule for the flat-top shrinkage. Bounds on the optimal bandwidth for noise models of varying complexity are also provided. In theoretical and numerical comparisons with alternative estimators, including the realized kernel, the two-scale realized kernel, and a bias-corrected pre-averaging estimator, the flat-top realized kernel enjoys a higher-order advantage in terms of bias reduction, in addition to good efficiency properties. The analysis is extended to jump-diffusions where the asymptotic properties of a flat-top realized kernel estimate of the total quadratic variation are established. Apart from a larger asymptotic variance, they are similar to the no-jump case. Finally, the estimators are used to design two classes of (medium) blocked realized kernels, which produce consistent, non-negative estimates of integrated variance. The blocked estimators are shown to have no loss either of asymptotic efficiency or in the rate of consistency relative to the flat-top realized kernels when jumps are absent. However, only the medium blocked realized kernels achieve the optimal rate of convergence under the jump alternative.

scholarly journals ADAPTIVE ESTIMATION OF FUNCTIONALS IN NONPARAMETRIC INSTRUMENTAL REGRESSION

2015 ◽  
Vol 32 (3) ◽  
pp. 612-654 ◽  
Author(s):  
Christoph Breunig ◽  
Jan Johannes
Keyword(s):  
Rate Of Convergence ◽  
Adaptive Estimation ◽  
Optimal Choice ◽  
Optimal Rate ◽  
Structural Function ◽  
Regularity Conditions ◽  
Adaptive Estimator ◽  
Optimal Rate Of Convergence ◽  
Pointwise Estimation ◽  
Additional Regularity

We consider the problem of estimating the valueℓ(ϕ) of a linear functional, where the structural functionϕmodels a nonparametric relationship in presence of instrumental variables. We propose a plug-in estimator which is based on a dimension reduction technique and additional thresholding. It is shown that this estimator is consistent and can attain the minimax optimal rate of convergence under additional regularity conditions. This, however, requires an optimal choice of the dimension parametermdepending on certain characteristics of the structural functionϕand the joint distribution of the regressor and the instrument, which are unknown in practice. We propose a fully data driven choice ofmwhich combines model selection and Lepski’s method. We show that the adaptive estimator attains the optimal rate of convergence up to a logarithmic factor. The theory in this paper is illustrated by considering classical smoothness assumptions and we discuss examples such as pointwise estimation or estimation of averages of the structural functionϕ.

scholarly journals The Virtual Element Method with curved edges

2019 ◽  
Vol 53 (2) ◽  
pp. 375-404 ◽  
Author(s):  
L. Beirão da Veiga ◽  
A. Russo ◽  
G. Vacca
Keyword(s):  
Rate Of Convergence ◽  
Two Dimensions ◽  
Optimal Rate ◽  
Virtual Element Method ◽  
Curved Boundary ◽  
Optimal Rate Of Convergence ◽  
Virtual Element ◽  
Curved Interface ◽  
Element Method

In this paper we initiate the investigation of Virtual Elements with curved faces. We consider the case of a fixed curved boundary in two dimensions, as it happens in the approximation of problems posed on a curved domain or with a curved interface. While an approximation of the domain with polygons leads, for degree of accuracy k≥2, to a sub-optimal rate of convergence, we show (both theoretically and numerically) that the proposed curved VEM lead to an optimal rate of convergence.

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