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Journal of Theoretical Probability

, Volume 31, Issue 3, pp 1273–1302 | Cite as

Weak Convergence of the Empirical Spectral Distribution of High-Dimensional Band Sample Covariance Matrices

  • Kamil Jurczak
Article
  • 37 Downloads

Abstract

In this article, we investigate high-dimensional band sample covariance matrices under the regime that the sample size n, the dimension p, and the bandwidth d tend simultaneously to infinity such that
$$\begin{aligned} n/p\rightarrow 0 \ \ \text {and} \ \ d/n\rightarrow y>0. \end{aligned}$$
It is shown that the empirical spectral distribution of those matrices converges weakly to a deterministic probability measure with probability 1. The limiting measure is characterized by its moments. Certain restricted compositions of natural numbers play a crucial role in the evaluation of the expected moments of the empirical spectral distribution.

Keywords

High-dimensional sample covariance matrices Empirical spectral distribution Strong convergence Weak convergence Method of moments Number of restricted compositions of a natural number 

Mathematics Subject Classification (2010)

60B20 

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany

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