Asymptotic inner product of correlated random vectors

Question

Suppose $\mathbf{x}$ and $\mathbf{y}$ are N-dimensional non-white complex random vectors independent of each other i.e., covariance matrices $\mathbf{C_{xx}}\neq\mathbf{I}$, $\mathbf{C_{yy}}\neq\mathbf{I}$, $\mathbf{C_{xy}}=\mathbf{0}$. Is there any results on asymptotic orthogonality $\mathbf{x}^H\mathbf{y}=\mathbf{u}^H\mathbf{C_{xx}}^{\frac{1}{2}}\mathbf{C_{yy}}^{\frac{1}{2}}\mathbf{v}=0$ where $\mathbf{u}$ and $\mathbf{v}$ are independent white vectors. Thanks.

Answer 1

I came up with this:

We have $\mathbf{x}^H\mathbf{y}=\mathbf{u}^H\mathbf{C_{xx}}^{\frac{1}{2}}\mathbf{C_{yy}}^{\frac{1}{2}}\mathbf{v}=tr\{\mathbf{C_{xx}}^{\frac{1}{2}}\mathbf{C_{yy}}^{\frac{1}{2}}\mathbf{v}\mathbf{u}^H\}\leq tr\{\mathbf{C_{xx}}^{\frac{1}{2}}\mathbf{C_{yy}}^{\frac{1}{2}}\}tr\{\mathbf{v}\mathbf{u}^H\}$ $=tr\{\mathbf{C_{xx}}^{\frac{1}{2}}\mathbf{C_{yy}}^{\frac{1}{2}}\}tr\{\mathbf{u}^H\mathbf{v}\}=0$.

Is this correct? Thanks.

Answer 2

I Think using the Theorem 3.4 in [1] (page 45), we can say:

$\boldsymbol x^H\boldsymbol y\xrightarrow{a.s.}tr(\boldsymbol C_{xx}^{\frac{1}{2}}\boldsymbol C_{yy}^{\frac{1}{2}})$

[1] Couillet R, Debbah M. Random matrix methods for wireless communications. Cambridge University Press; 2011 Sep 29.