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THE OUTLIERS AMONG THE SINGULAR VALUES OF LARGE RECTANGULAR RANDOM MATRICES WITH ADDITIVE FIXED RANK DEFORMATION

Abstract : Consider the matrix Σn = n −1/2 XnD 1/2 n + Pn where the matrix Xn ∈ C N×n has Gaussian standard independent elements, Dn is a deter-ministic diagonal nonnegative matrix, and Pn is a deterministic matrix with fixed rank. Under some known conditions, the spectral measures of ΣnΣ * n and n −1 XnDnX * n both converge towards a compactly supported probability measure µ as N, n → ∞ with N/n → c > 0. In this paper, it is proved that finitely many eigenvalues of ΣnΣ * n may stay away from the support of µ in the large dimensional regime. The existence and locations of these outliers in any connected component of R − supp(µ) are studied. The fluctuations of the largest outliers of ΣnΣ * n are also analyzed. The results find applications in the fields of signal processing and radio communications.
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Submitted on : Tuesday, December 30, 2014 - 10:17:57 AM
Last modification on : Wednesday, June 24, 2020 - 4:19:00 PM
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  • HAL Id : hal-01098923, version 1
  • ARXIV : 1207.0471

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Francois Chapon, Romain Couillet, Walid Hachem, Xavier Mestre. THE OUTLIERS AMONG THE SINGULAR VALUES OF LARGE RECTANGULAR RANDOM MATRICES WITH ADDITIVE FIXED RANK DEFORMATION. Markov Processes and Related Fields, Polymath, 2014, 20 (2), pp.183-228. ⟨hal-01098923⟩

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