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

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  • Kamil Jurczak

    (Ruhr-Universität Bochum)

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 $$\begin{aligned} n/p\rightarrow 0 \ \ \text {and} \ \ d/n\rightarrow y>0. \end{aligned}$$ n / p → 0 and d / n → y > 0 . 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.

Suggested Citation

  • Kamil Jurczak, 2018. "Weak Convergence of the Empirical Spectral Distribution of High-Dimensional Band Sample Covariance Matrices," Journal of Theoretical Probability, Springer, vol. 31(3), pages 1273-1302, September.
    • Handle: RePEc:spr:jotpro:v:31:y:2018:i:3:d:10.1007_s10959-017-0751-7
    DOI: 10.1007/s10959-017-0751-7
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    References listed on IDEAS

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    1. Lam, Clifford & Fan, Jianqing, 2009. "Sparsistency and rates of convergence in large covariance matrix estimation," LSE Research Online Documents on Economics 31540, London School of Economics and Political Science, LSE Library.
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