  Random Fisher matrices arise naturally in multivariate statistical analysis
and understanding the properties of its eigenvalues is of primary importance
for many hypothesis testing problems like testing the equality between two
multivariate population covariance matrices, or testing the independence
between sub-groups of a multivariate random vector. This paper is concerned
with the properties of a large-dimensional Fisher matrix when the dimension of
the population is proportionally large compared to the sample size. Most of
existing works on Fisher matrices deal with a particular Fisher matrix where
populations have i.i.d components so that the population covariance matrices
are all identity. In this paper, we consider general Fisher matrices with
arbitrary population covariance matrices. The first main result of the paper
establishes the limiting distribution of the eigenvalues of a Fisher matrix
while in a second main result, we provide a central limit theorem for a wide
class of functionals of its eigenvalues. Some applications of these results are
also proposed for testing hypotheses on high-dimensional covariance matrices.
