On the Distribution of the `th Largest Eigenvalue of Spiked Complex Wishart Matrices

The study of the statistical distribution of the eigenvalues of Wishart matrices finds application in many fields of physics and engineering. Here, we consider a special case of finite dimensions correlated complex central Wishart matrices, characterized by the fact that the covariance matrix has all eigenvalues equal, except for one which is the largest. Starting from the knowledge of the joint probability distribution function (p.d.f.) of this kind of Wishart matrices, we focus on the evaluation of a tractable form for the distribution of each individual eigenvalue. In particular, we derive an expression for the p.d.f. of the `th largest eigenvalue as a sum of terms of the type x^? exp(-x?), which allows us to write a large class of statistical averages involving functions of eigenvalues in closed form.