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

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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.

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Authors and Affiliations

  1. Fakultät für Mathematik, Ruhr-Universität Bochum, 44780, Bochum, Germany

    Kamil Jurczak

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  1. Kamil Jurczak
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Correspondence to Kamil Jurczak.

Additional information

Supported by the DFG Research Unit 1735, RO 3766/3-1 and DE 502/26-2.

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Jurczak, K. Weak Convergence of the Empirical Spectral Distribution of High-Dimensional Band Sample Covariance Matrices. J Theor Probab 31, 1273–1302 (2018). https://doi.org/10.1007/s10959-017-0751-7

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  • DOI: https://doi.org/10.1007/s10959-017-0751-7

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