Large complex correlated Wishart matrices: the Pearcey kernel and expansion at the hard edge

Abstract : We study the eigenvalue behaviour of large complex correlated Wishart matrices near an interior point of the limiting spectrum where the density vanishes (cusp point), and refine the existing results at the hard edge as well. More precisely, under mild assumptions for the population covariance matrix, we show that the limiting density vanishes at generic cusp points like a cube root, and that the local eigenvalue behaviour is described by means of the Pearcey kernel if an extra decay assumption is satisfied. As for the hard edge, we show that the density blows up like an inverse square root at the origin. Moreover, we provide an explicit formula for the 1/N correction term for the fluctuation of the smallest random eigenvalue.
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Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2016, 21 (1), pp.1-36. 〈10.1214/15-EJP4441〉
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Contributeur : Jamal Najim <>
Soumis le : mardi 10 juillet 2018 - 15:49:15
Dernière modification le : jeudi 12 juillet 2018 - 01:28:12

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Walid Hachem, Adrien Hardy, Jamal Najim. Large complex correlated Wishart matrices: the Pearcey kernel and expansion at the hard edge. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2016, 21 (1), pp.1-36. 〈10.1214/15-EJP4441〉. 〈hal-01834495〉

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