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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”) $S = M^{-1}(X - \bar X)(X -\bar X)^T$, where $X$ are rectangular $N \times M$ sample matrices whose columns are observation vectors distributed as $N(0,\Sigma)$, and $\bar X$ are sample mean vectors. It is assumed that $N, M \rightarrow\infty$ and $M/N \rightarrow\gamma^2 > 1$. In this asymptotics, when $\Sigma$ are unity matrices, {\it V. A. Marchenko} and {\it 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 $(1 + 1/\gamma)^2$. For this case also, the largest eigenvalue $\lambda_1$ of the limiting distribution was found by {\it P. J. Forrester} [Nucl. Phys. B 402, No. 3, 709--728 (1993; Zbl 1043.82538)] $$\Bbb P(aM^{2/3}(\lambda_1- (1 + 1/\gamma)^2< x) \rightarrow F(x),\tag1$$ where $a$ depends only on $\gamma$. The authors find the limiting distribution of the largest eigenvalue $\lambda_1$ for complex matrices $S$ when there is a finite number of eigenvalues of $\Sigma$ different from 1. Let $r \leq N$ and the eigenvalues $l_1 \geq l_2 \geq\dots l_N$ of $\Sigma$ be such that $l_{r+1} = l_{r+2} = \dots = l_N.$ Theorem 1 states that (i) If $k\leq r$, $l_1=l_2=\dots=l_k=1+1/\gamma$ and all $\lambda_j$, $j=k+1,k+2\dots,r$, are located on a compact within $(0,1 + 1/\gamma)$, then the largest eigenvalue $\lambda_1$ of $S$ has the same limiting distribution (1); (ii) If for some $k\leq r$, the eigenvalues $l_1=l_2=\dots=l_k$ are located on a compact within $(1 + 1/\gamma,\infty)$ and all eigenvalues $l_j$, $j=k+1,k+2\dots,r$, are on a compact within $(0,l_1)$, then $$\Bbb P(aM^{1/2}(\lambda_1-c)\leq x)\to G_k(x),$$ where $a > 0$ and $c < l_1$ depend on $\gamma$ and $l_1$, and $G_k(x)$ is the distribution function for the largest eigenvalue of $k\times k$ random matrix when $\Sigma$ 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:
 15B52 Random matrices 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 62E20 Asymptotic distribution theory in statistics 15A18 Eigenvalues, singular values, and eigenvectors 60E05 General theory of probability distributions 82B26 Phase transitions (general) 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60K25 Queueing theory
Full Text:
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