*(English)*Zbl 1086.15022

The largest eigenvalue distribution is investigated for $N$-dimensional random Gram matrices (“covariance matrices”) $S={M}^{-1}(X-\overline{X}){(X-\overline{X})}^{T}$, where $X$ are rectangular $N\times M$ sample matrices whose columns are observation vectors distributed as $N(0,{\Sigma})$, and $\overline{X}$ are sample mean vectors. It is assumed that $N,M\to \infty $ and $M/N\to {\gamma}^{2}>1$. In this asymptotics, when ${\Sigma}$ 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 ${(1+1/\gamma )}^{2}$. For this case also, the largest eigenvalue ${\lambda}_{1}$ of the limiting distribution was found by *P. J. Forrester* [Nucl. Phys. B 402, No. 3, 709–728 (1993; Zbl 1043.82538)]

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 $r\le N$ and the eigenvalues ${l}_{1}\ge {l}_{2}\ge \cdots {l}_{N}$ of ${\Sigma}$ be such that ${l}_{r+1}={l}_{r+2}=\cdots ={l}_{N}\xb7$

Theorem 1 states that

(i) If $k\le r$, ${l}_{1}={l}_{2}=\cdots ={l}_{k}=1+1/\gamma $ and all ${\lambda}_{j}$, $j=k+1,k+2\cdots ,r$, 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 $k\le r$, the eigenvalues ${l}_{1}={l}_{2}=\cdots ={l}_{k}$ are located on a compact within $(1+1/\gamma ,\infty )$ and all eigenvalues ${l}_{j}$, $j=k+1,k+2\cdots ,r$, are on a compact within $(0,{l}_{1})$, then

where $a>0$ and $c<{l}_{1}$ depend on $\gamma $ and ${l}_{1}$, and ${G}_{k}\left(x\right)$ 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:

15A52 | Random matrices (MSC2000) |

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 |