# Asymptotic analysis for extreme eigenvalues of principal minors of random matrices

T. Tony Cai, Tiefeng Jiang, L. I. Xiaoou

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3 Scopus citations

## Abstract

Consider a standard white Wishart matrix with parameters n and p. Motivated by applications in high-dimensional statistics and signal processing, we perform asymptotic analysis on the maxima and minima of the eigenvalues of all the m × m principal minors, under the asymptotic regime that n, p, m go to infinity. Asymptotic results concerning extreme eigenvalues of principal minors of real Wigner matrices are also obtained. In addition, we discuss an application of the theoretical results to the construction of compressed sensing matrices, which provides insights to compressed sensing in signal processing and high-dimensional linear regression in statistics.

Original language English (US) 2953-2990 38 Annals of Applied Probability 31 6 https://doi.org/10.1214/21-AAP1668 Published - Dec 2021

### Bibliographical note

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PROOF OF LEMMA 12. (i) =⇒ (ii): Easily, E[eαZp] ≤ E[eαZp1{Zp≥δ}] + eαδ. Taking lim supp→∞ on both sides and then letting δ ↓ 0, we obtain lim supp→∞ E[eαZp] ≤ 1. On the otherside,liminfp→∞E[eαZp]≥1sinceZp≥0.Hence,limp→∞E[eαZp]=1. (ii) =⇒ (i):Foreachβ>0,weknow1{Zp≥δ}≤eβ(Zp−δ).Thus, [αZ][αZ+β(Z−δ)]−βδ[(α+β)Z] Ee p1{Zp≥δ} ≤E e p p =e E e p . Taking lim supp→∞ on both sides and then letting β → ∞, we obtain limp→∞E[eαZp 1{Zp≥δ}] = 0. (ii) =⇒ (iii): First, E[Zpα] = α0∫∞xα−1P(Zp ≥ x)dx. By the Markov inequality, P(Zp≥x)≤e−βxEeβZpforallx>0andβ>0.ItfollowsthatE[Zα]≤α(EeβZp)× ∫ ∞ α−1 −βx αΓ(α) βZpp 0xedx=βαEeforallβ>0.Theconclusionthenfollowsbyfirstletting p → ∞ and then sending β → ∞. (iii) =⇒ (iv): This is a direct consequence of the Chebyshev inequality and the equality limp→∞E(Zp) = 0. (iii) =⇒ (v): Let α = 2 in (iii), then lim supp Var(Zp) ≤ limp→∞ E(Zp2) = 0. □ Funding. The research of Tony Cai was supported in part by NSF Grants DMS-1712735 and DMS-2015259 and NIH Grants R01-GM129781 and R01-GM123056. Tiefeng Jiang’s research was supported by NSF Grants DMS-1406279 and DMS-1916014. Xiaoou Li was partially supported by NSF Grant DMS-1712657.

© Institute of Mathematical Statistics, 2021

## Keywords

• Extremal eigenvalues
• Maximum of random variables
• Minimum of random variables
• Random matrix

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