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On eigenvalues of the Brownian sheet matrix

Author

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  • Song, Jian
  • Xiao, Yimin
  • Yuan, Wangjun

Abstract

We derive a system of stochastic partial differential equations satisfied by the eigenvalues of the symmetric matrix whose entries are the Brownian sheets. We prove that the sequence Ld(s,t),(s,t)∈[0,S]×[0,T]d∈N of empirical spectral measures of the rescaled matrices is tight on C([0,S]×[0,T],P(R)) and hence is convergent as d goes to infinity by Wigner’s semicircle law. We also obtain PDEs which are satisfied by the high-dimensional limiting measure.

Suggested Citation

  • Song, Jian & Xiao, Yimin & Yuan, Wangjun, 2023. "On eigenvalues of the Brownian sheet matrix," Stochastic Processes and their Applications, Elsevier, vol. 166(C).
    • Handle: RePEc:eee:spapps:v:166:y:2023:i:c:s030441492300203x
    DOI: 10.1016/j.spa.2023.104231
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    References listed on IDEAS

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    1. Ivan Nourdin & Murad S. Taqqu, 2014. "Central and Non-central Limit Theorems in a Free Probability Setting," Journal of Theoretical Probability, Springer, vol. 27(1), pages 220-248, March.
    2. Juan Carlos Pardo & José-Luis Pérez & Victor Pérez-Abreu, 2016. "A Random Matrix Approximation for the Non-commutative Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 29(4), pages 1581-1598, December.
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