Consider I have an $M\times 1$ vector $\boldsymbol x$ whose elements are i.i.d. random variables. Based on Theorem 3.4 in [1], as $M\to\infty$, we have :
$\boldsymbol{x}^H\boldsymbol{A}\boldsymbol{x}\xrightarrow{a.s.}tr(\boldsymbol{A})$
Now, consider that I form another vector $\boldsymbol{y}=\boldsymbol{\theta}\boldsymbol{x}$. Can I conclude that as $M\to\infty$,
$\boldsymbol{y}^H\boldsymbol{A}\boldsymbol{y}\xrightarrow{a.s.}tr(\boldsymbol{\theta A\theta})$
?
Thanks in advance.
[1] Couillet R, Debbah M. Random matrix methods for wireless communications. Cambridge University Press; 2011 Sep 29.