  Consider the quadratic form $\beta = {\bf y}^* ({\bf YY}^* + \rho {\bf
I})^{-1} {\bf y}$ where $\rho$ is a positive number, where ${\bf y}$ is a
random vector and ${\bf Y}$ is a $N \times K$ random matrix both having
independent elements with different variances, and where ${\bf y}$ and ${\bf
Y}$ are independent. Such quadratic forms represent the Signal to Noise Ratio
at the output of the linear Wiener receiver for multi dimensional signals
frequently encountered in wireless communications and in array processing.
Using well known results of Random Matrix Theory, the quadratic form $\beta$
can be approximated with a known deterministic real number $\bar\beta_K$ in the
asymptotic regime where $K\to\infty$ and $K/N \to \alpha > 0$. This paper
addresses the problem of convergence of $\beta$. More specifically, it is shown
here that $\sqrt{K}(\beta - \bar\beta_K)$ behaves for large $K$ like a Gaussian
random variable which variance is provided.
