The largest eigenvalue distribution is investigated for -dimensional random Gram matrices (“covariance matrices”) , where are rectangular sample matrices whose columns are observation vectors distributed as , and are sample mean vectors. It is assumed that and . In this asymptotics, when 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 ; its upper bound was found to be . For this case also, the largest eigenvalue of the limiting distribution was found by P. J. Forrester [Nucl. Phys. B 402, No. 3, 709–728 (1993; Zbl 1043.82538)]
where depends only on .
The authors find the limiting distribution of the largest eigenvalue for complex matrices when there is a finite number of eigenvalues of different from 1. Let and the eigenvalues of be such that
Theorem 1 states that
(i) If , and all , , are located on a compact within , then the largest eigenvalue of has the same limiting distribution (1);
(ii) If for some , the eigenvalues are located on a compact within and all eigenvalues , , are on a compact within , then
where and depend on and , and is the distribution function for the largest eigenvalue of random matrix when 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 customers in the line of tellers under a fixed order of serving.