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Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. (English) Zbl 1086.15022

The largest eigenvalue distribution is investigated for N-dimensional random Gram matrices (“covariance matrices”) S=M -1 (X-X ¯)(X-X ¯) T , where X are rectangular N×M sample matrices whose columns are observation vectors distributed as N(0,Σ), and X ¯ are sample mean vectors. It is assumed that N,M and M/Nγ 2 >1. In this asymptotics, when Σ are unity matrices, V. A. Marchenko and L. A. Pastur [Math. USSR, Sb. 1, 457–483 (1967; Zbl 0162.22501)] derived well-known limiting distribution density of eigenvalues of S; its upper bound was found to be (1+1/γ) 2 . For this case also, the largest eigenvalue λ 1 of the limiting distribution was found by P. J. Forrester [Nucl. Phys. B 402, No. 3, 709–728 (1993; Zbl 1043.82538)]

(aM 2/3 (λ 1 -(1+1/γ) 2 <x)F(x),(1)

where a depends only on γ.

The authors find the limiting distribution of the largest eigenvalue λ 1 for complex matrices S when there is a finite number of eigenvalues of Σ different from 1. Let rN and the eigenvalues l 1 l 2 l N of Σ be such that l r+1 =l r+2 ==l N ·

Theorem 1 states that

(i) If kr, l 1 =l 2 ==l k =1+1/γ and all λ j , j=k+1,k+2,r, are located on a compact within (0,1+1/γ), then the largest eigenvalue λ 1 of S has the same limiting distribution (1);

(ii) If for some kr, the eigenvalues l 1 =l 2 ==l k are located on a compact within (1+1/γ,) and all eigenvalues l j , j=k+1,k+2,r, are on a compact within (0,l 1 ), then

(aM 1/2 (λ 1 -c)x)G k (x),

where a>0 and c<l 1 depend on γ and l 1 , and G k (x) is the distribution function for the largest eigenvalue of k×k random matrix when Σ is unity.

Theorem 2 establishes the limiting distribution (1) in an intermediate case. Two applications are considered: finding the last passage time in a percolation problem and a queueing model of a sequential service of M customers in the line of N tellers under a fixed order of serving.


MSC:
15A52Random matrices (MSC2000)
60B12Limit theorems for vector-valued random variables (infinite-dimensional case)
62E20Asymptotic distribution theory in statistics
15A18Eigenvalues, singular values, and eigenvectors
60E05General theory of probability distributions
82B26Phase transitions (general)
60K35Interacting random processes; statistical mechanics type models; percolation theory
60K25Queueing theory