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

2016; Institute of Mathematical Statistics; Volume: 21; Issue: none Linguagem: Inglês

10.1214/15-ejp4441

1083-6489

Walid Hachem, Adrien Hardy, Jamal Najım,

Statistical Methods and Bayesian Inference

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.