scholarly journals On the universality of spectral limit for random matrices with martingale differences entries

2015 ◽  
Vol 04 (01) ◽  
pp. 1550003 ◽  
Author(s):  
F. Merlevède ◽  
C. Peligrad ◽  
M. Peligrad
Keyword(s):  
Random Matrices ◽  
Infinite Order ◽  
Regularity Conditions ◽  
Martingale Differences ◽  
Full Extension ◽  
Semicircle Law ◽  
Convergence Results ◽  
Nonlinear Autoregressive ◽  
Martingale Case ◽  
Variance Structure

For a class of symmetric random matrices whose entries are martingale differences adapted to an increasing filtration, we prove that under a Lindeberg-like condition, the empirical spectral distribution behaves asymptotically similarly to a corresponding matrix with independent centered Gaussian entries having the same variances. Under a slightly reinforced condition, the approximation holds in the almost sure sense. We also point out several sufficient regularity conditions imposed to the variance structure for convergence to the semicircle law or the Marchenko–Pastur law and other convergence results. In the stationary case, we obtain a full extension from the i.i.d. case to the martingale case of the convergence to the semicircle law as well as to the Marchenko–Pastur one. Our results are well adapted to study several examples including nonlinear autoregressive conditional heteroscedastic random fields of infinite order.

Limit theorems for point processes generated in a general branching process

1981 ◽  
Vol 13 (04) ◽  
pp. 650-668 ◽  
Author(s):  
Martin Härnqvist
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Branching Processes ◽  
Special Problem ◽  
Convergence Theory ◽  
Regularity Conditions ◽  
Convergence Results ◽  
Mixed Poisson Process ◽  
Proper Scaling ◽  
Extra Point

With the general convergence theory for branching processes as basis a special problem is studied. An extra point process of events during life is assigned to each realised individual, and the behaviour of the superposition of such point processes in action is studied as the population grows. With the proper scaling and under some regularity conditions the superposition is shown to converge in distribution to a Poisson process. Another scaling gives rise to a mixed Poisson process as limit. Established weak convergence techniques for point processes are applied, together with some recent strong convergence results for branching processes.

scholarly journals EIGENVALUE VARIANCE BOUNDS FOR WIGNER AND COVARIANCE RANDOM MATRICES

2012 ◽  
Vol 01 (03) ◽  
pp. 1250007 ◽  
Author(s):  
S. DALLAPORTA
Keyword(s):  
Random Matrices ◽  
Spectral Measure ◽  
Wasserstein Distance ◽  
Covariance Matrices ◽  
Finite Range ◽  
Variance Bounds ◽  
Semicircle Law ◽  
Wigner Matrices ◽  
Wigner Random Matrices

This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example, which needs to be investigated first, the main bounds are extended to families of Hermitian Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erdös, Yau and Yin. The case of real Wigner matrices is obtained from interlacing formulas. As an application, bounds on the expected 2-Wasserstein distance between the empirical spectral measure and the semicircle law are derived. Similar results are available for random covariance matrices.

scholarly journals Random Matrices From Linear Codes and Wigner’s Semicircle Law

2019 ◽  
Vol 65 (10) ◽  
pp. 6001-6009
Author(s):  
Chin Hei Chan ◽  
Enoch Kung ◽  
Maosheng Xiong
Keyword(s):  
Random Matrices ◽  
Linear Codes ◽  
Semicircle Law

ASYMPTOTIC THEORY FOR SPECTRAL DENSITY ESTIMATES OF GENERAL MULTIVARIATE TIME SERIES

2017 ◽  
Vol 34 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Wei Biao Wu ◽  
Paolo Zaffaroni
Keyword(s):  
Time Series ◽  
Spectral Density ◽  
Multivariate Time Series ◽  
Rates Of Convergence ◽  
Stationary Time Series ◽  
Regularity Conditions ◽  
Dynamic Factor ◽  
Asymptotic Results ◽  
Density Estimates ◽  
Convergence Results

We derive uniform convergence results of lag-window spectral density estimates for a general class of multivariate stationary processes represented by an arbitrary measurable function of iid innovations. Optimal rates of convergence, that hold as both the time series and the cross section dimensions diverge, are obtained under mild and easily verifiable conditions. Our theory complements earlier results, most of which are univariate, which primarily concern in-probability, weak or distributional convergence, yet under a much stronger set of regularity conditions, such as linearity in iid innovations. Based on cross spectral density functions, we then propose a new test for independence between two stationary time series. We also explain the extent to which our results provide the foundation to derive the double asymptotic results for estimation of generalized dynamic factor models.

scholarly journals Weighted multiple ergodic averages and correlation sequences

2016 ◽  
Vol 38 (1) ◽  
pp. 81-142 ◽  
Author(s):  
NIKOS FRANTZIKINAKIS ◽  
BERNARD HOST
Keyword(s):  
Bounded Sequence ◽  
Regularity Conditions ◽  
Uniform Density ◽  
Ergodic Averages ◽  
Multiple Correlation ◽  
Mean Convergence ◽  
Integer Polynomials ◽  
Several Variables ◽  
Convergence Results ◽  
Correlation Sequences

We study mean convergence results for weighted multiple ergodic averages defined by commuting transformations with iterates given by integer polynomials in several variables. Roughly speaking, we prove that a bounded sequence is a good universal weight for mean convergence of such averages if and only if the average of this sequence times any nilsequence converges. Two decomposition results of independent interest play key roles in the proof. The first states that every bounded sequence in several variables satisfying some regularity conditions is a sum of a nilsequence and a sequence that has small uniformity norm (this generalizes a result of the second author and Kra); and the second states that every multiple correlation sequence in several variables is a sum of a nilsequence and a sequence that is small in uniform density (this generalizes a result of the first author). Furthermore, we use these results in order to establish mean convergence and recurrence results for a variety of sequences of dynamical and arithmetic origin and give some combinatorial implications.

Formal Theory

2011 ◽  
Author(s):  
Charles Fefferman ◽  
C. Robin Graham
Keyword(s):  
Normal Form ◽  
Ricci Curvature ◽  
Infinite Order ◽  
Formal Theory ◽  
Formal Series ◽  
Ricci Flat ◽  
Convergence Results ◽  
Real Analytic

This chapter presents proof of Theorem 2.9 for n > 2. It further notes that similar arguments using the form of the perturbation formulae (3.32) for the Ricci curvature show that the metrics constructed in Theorems 3.7, 3.9 and 3.10 are the only formal expansions of metrics for ρ‎ > 0 or ρ‎ < 0 involving positive powers of ¦ ρ‎ r ρ‎ and log ¦ ρ‎ r ρ‎, which are homogeneous of degree 2, Ricci-flat to infinite order, and in normal form. Convergence of formal series determined by Fuchsian problems such as these in the case of real-analytic data has been considered by several authors. In particular, results of [BaoG] can be applied to establish the convergence of the series occurring in Theorems 3.7 and 3.9 (and also in Theorem 3.10 if the obstruction tensor vanishes) if g and h are real-analytic. Convergence results including also the case when log terms occur in Theorem 3.10 are contained in [K].

Wigner matrices

2018 ◽  
Author(s):  
Peter Forrester
Keyword(s):  
Simple Model ◽  
Random Matrices ◽  
Symmetric Matrices ◽  
Semicircle Law ◽  
Wigner Matrices ◽  
Global Properties ◽  
Wigner Random Matrices ◽  
Local Properties ◽  
Concentration Properties ◽  
Adjoint Operators

This article reviews some of the important results in the study of the eigenvalues and the eigenvectors of Wigner random matrices, that is. random Hermitian (or real symmetric) matrices with iid entries. It first provides an overview of the Wigner matrices, introduced in the 1950s by Wigner as a very simple model of random matrices to approximate generic self-adjoint operators. It then considers the global properties of the spectrum of Wigner matrices, focusing on convergence to the semicircle law, fluctuations around the semicircle law, deviations and concentration properties, and the delocalization of the eigenvectors. It also describes local properties in the bulk and at the edge before concluding with a brief analysis of the known universality results showing how much the behaviour of the spectrum is insensitive to the distribution of the entries.

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