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Fluctuations of linear statistics of half-heavy-tailed random matrices

Author

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  • Benaych-Georges, Florent
  • Maltsev, Anna

Abstract

In this paper, we consider a Wigner matrix A with entries whose cumulative distribution decays as x−α with 2<α<4 for large x. We are interested in the fluctuations of the linear statistics N−1Trφ(A), for some nice test functions φ. The behavior of such fluctuations has been understood for both heavy-tailed matrices (i.e. α<2) in Benaych-Georges (2014) and light-tailed matrices (i.e. α>4) in Bai and Silverstein (2009). This paper fills in the gap of understanding it for 2<α<4. We find that while linear spectral statistics for heavy-tailed matrices have fluctuations of order N−1/2 and those for light-tailed matrices have fluctuations of order N−1, the linear spectral statistics for half-heavy-tailed matrices exhibit an intermediate α-dependent order of N−α/4.

Suggested Citation

  • Benaych-Georges, Florent & Maltsev, Anna, 2016. "Fluctuations of linear statistics of half-heavy-tailed random matrices," Stochastic Processes and their Applications, Elsevier, vol. 126(11), pages 3331-3352.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:11:p:3331-3352
    DOI: 10.1016/j.spa.2016.04.030
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