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Smallest singular value and limit eigenvalue distribution of a class of non-Hermitian random matrices with statistical application

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  • Bose, Arup
  • Hachem, Walid

Abstract

Suppose X is an N×n complex matrix whose entries are centered, independent, and identically distributed random variables with variance 1∕n and whose fourth moment is of order O(n−2). Suppose A is a deterministic matrix whose smallest and largest singular values are bounded below and above respectively, and z≠0 is a complex number. First we consider the matrix XAX∗−z, and obtain asymptotic probability bounds for its smallest singular value when N and n diverge to infinity and N∕n→γ,0<γ<∞. Then we consider the special case where A=J=[1i−j=1modn] is a circulant matrix. Using the above result, we show that the limit spectral distribution of XJX∗ exists when N∕n→γ,0<γ<∞ and describe the limit explicitly. Assuming that X represents a ℂN-valued time series which is observed over a time window of length n, the matrix XJX∗ represents the one-step sample autocovariance matrix of this time series. A whiteness test against an MA correlation model for this time series is introduced based on the above limit result. Numerical simulations show the excellent performance of this test.

Suggested Citation

  • Bose, Arup & Hachem, Walid, 2020. "Smallest singular value and limit eigenvalue distribution of a class of non-Hermitian random matrices with statistical application," Journal of Multivariate Analysis, Elsevier, vol. 178(C).
  • Handle: RePEc:eee:jmvana:v:178:y:2020:i:c:s0047259x19305688
    DOI: 10.1016/j.jmva.2020.104623
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    References listed on IDEAS

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    1. Weiming Li & Zeng Li & Jianfeng Yao, 2018. "Joint Central Limit Theorem for Eigenvalue Statistics from Several Dependent Large Dimensional Sample Covariance Matrices with Application," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 45(3), pages 699-728, September.
    2. Monika Bhattacharjee & Arup Bose, 2014. "Estimation Of Autocovariance Matrices For Infinite Dimensional Vector Linear Process," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(3), pages 262-281, May.
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    Cited by:

    1. Sanders, Jaron & Van Werde, Alexander, 2023. "Singular value distribution of dense random matrices with block Markovian dependence," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 453-504.

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