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

Abstract : In this paper, we consider a Wigner matrix $A$ with entries whose cumulative distribution decays as $x^{-\alpha}$ with $2<\alpha<4$ for large $x$. We prove that the fluctuations of the linear statistics $N^{-1}\operatorname{Tr} \varphi(A)$, for some nice test functions $\varphi$, have order $N^{-\alpha/4}$. The behavior of such fluctuations has been understood for both heavy-tailed matrices (i.e. $\alpha < 2$) and light-tailed matrices (i.e. $\alpha > 4$). This paper fills in the gap of understanding for $2 < \alpha < 4$. 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 $\alpha$-dependent order of $N^{-\alpha/4}$.
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Stochastic Processes and their Applications, Elsevier, 2016, <10.1016/j.spa.2016.04.030>
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https://hal.archives-ouvertes.fr/hal-01076536
Contributeur : Florent Benaych-Georges <>
Soumis le : mercredi 22 octobre 2014 - 14:07:32
Dernière modification le : mardi 3 janvier 2017 - 14:22:00

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Florent Benaych-Georges, Anna Maltsev. Fluctuations of linear statistics of half-heavy-tailed random matrices. Stochastic Processes and their Applications, Elsevier, 2016, <10.1016/j.spa.2016.04.030>. <hal-01076536>

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