Shrinkage estimators of covariance are an important tool in modern applied
and theoretical statistics. They play a key role in regularized estimation
problems, such as ridge regression (aka Tykhonov regularization), regularized
discriminant analysis and a variety of optimization problems.

In this paper, we bring to bear the tools of random matrix theory to
understand their behavior, and in particular, that of quadratic forms involving
inverses of those estimators, which are important in practice.

We place ourselves in the setting of high-dimensional statistical inference
where the number of variables $p$ in a dataset of interest is of the same order
of magnitude as the number of observations $n$. We consider the spectrum of
certain kernel random matrices, in particular $n\times n$ matrices whose
$(i,j)$th entry is $f(X_i'X_j/p)$ or $f(\Vert X_i-X_j\Vert^2/p)$ where $p$ is
the dimension of the data, and $X_i$ are independent data vectors. Here $f$ is
assumed to be a locally smooth function.