Phase transition for the smallest eigenvalue of covariance matrices
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In this paper, we study the smallest non-zero eigenvalue of the sample covariance matrices 𝒮(Y)=YY^*, where Y=(y_ij) is an M× N matrix with iid mean 0 variance N^-1 entries. We prove a phase transition for its distribution, induced by the fatness of the tail of y_ij's. More specifically, we assume that y_ij is symmetrically distributed with tail probability ℙ(|√(N)y_ij|≥ x)∼ x^-α when x→∞, for some α∈ (2,4). We show the following conclusions: (i). When α>8/3, the smallest eigenvalue follows the Tracy-Widom law on scale N^-2/3; (ii). When 2<α<8/3, the smallest eigenvalue follows the Gaussian law on scale N^-α/4; (iii). When α=8/3, the distribution is given by an interpolation between Tracy-Widom and Gaussian; (iv). In case α≤10/3, in addition to the left edge of the MP law, a deterministic shift of order N^1-α/2 shall be subtracted from the smallest eigenvalue, in both the Tracy-Widom law and the Gaussian law. Overall speaking, our proof strategy is inspired by <cit.> which is originally done for the bulk regime of the Lévy Wigner matrices. In addition to various technical complications arising from the bulk-to-edge extension, two ingredients are needed for our derivation: an intermediate left edge local law based on a simple but effective matrix minor argument, and a mesoscopic CLT for the linear spectral statistic with asymptotic expansion for its expectation.
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