渐进推导中常用的一些结论

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STAR-RIS Enhanced Joint Physical Layer Security and Covert Communications for Multi-antenna mmWave Systems文章末尾的一个推导。

$$\begin{array}{l}{\lim }_{M\to \infty }\frac{{\Vert \mathbf{\Phi }\left({\mathbf{w}}_k\otimes {\mathbf{\Theta }}_{\mathrm{r}}\right){\mathbf{\Omega }}_{\mathrm{rw}}^H{\mathbf{g}}^{\ast }\Vert }_2^2}{M}\\ ={\lim }_{M\to \infty }\frac{\mathrm{Tr}\left({\mathbf{g}}^T{\mathbf{\Omega }}_{\mathrm{rw}}{\left({\mathbf{w}}_k\otimes {\mathbf{\Theta }}_{\mathrm{r}}\right)}^H{\mathbf{\Phi }}^H\mathbf{\Phi }\left({\mathbf{w}}_k\otimes {\mathbf{\Theta }}_{\mathrm{r}}\right){\mathbf{\Omega }}_{\mathrm{rw}}^H{\mathbf{g}}^{\ast }\right)}{M}\\ \stackrel{(a)}{\to }\frac{\mathrm{Tr}\left({\mathbf{\Omega }}_{\mathrm{rw}}^H{\mathbf{\Omega }}_{\mathrm{rw}}{\left({\mathbf{w}}_k\otimes {\mathbf{\Theta }}_{\mathrm{r}}\right)}^H{\mathbf{\Phi }}^H\mathbf{\Phi }\left({\mathbf{w}}_k\otimes {\mathbf{\Theta }}_{\mathrm{r}}\right)\right)}{M}\\ \stackrel{(b)}{=}\frac{\mathrm{Tr}\left({\sum }_{l=1}^L\left({\mathbf{w}}_k^H{\mathbf{\Psi }}_{\mathrm{BR}}^l{\mathbf{w}}_k\right)\otimes \left({\mathbf{\Omega }}_{\mathrm{rw}}^H{\mathbf{\Omega }}_{\mathrm{rw}}{\mathbf{\Theta }}_{\mathrm{r}}^H{\hat{\mathbf{\Psi }}}_{\mathrm{BR}}^l{\mathbf{\Theta }}_{\mathrm{r}}\right)\right)}{M}\\ \stackrel{(c)}{=}\frac{{\sum }_{l=1}^L\left({\mathbf{w}}_k^H{\mathbf{\Psi }}_{\mathrm{BR}}^l{\mathbf{w}}_k\right)\mathrm{Tr}\left({\mathbf{\Omega }}_{\mathrm{rw}}^H{\mathbf{\Omega }}_{\mathrm{rw}}{\mathbf{\Theta }}_{\mathrm{r}}^H{\hat{\mathbf{\Psi }}}_{\mathrm{BR}}^l{\mathbf{\Theta }}_{\mathrm{r}}\right)}{M}\\ =\frac{{\sum }_{l=1}^L\left({\mathbf{w}}_k^H{\mathbf{\Psi }}_{\mathrm{BR}}^l{\mathbf{w}}_k\right)\left({\mathbf{\vartheta }}_{\mathrm{r}}^T{\mathbf{\Xi }}^T\left({\left({\hat{\Psi }}_{\mathrm{BR}}^l\right)}^T\otimes \left({\mathbf{\Omega }}_{\mathrm{rw}}^H{\mathbf{\Omega }}_{\mathrm{rw}}\right)\right)\mathbf{\Xi }{\mathbf{\vartheta }}_{\mathrm{r}}^{\ast }\right)}{M},\end{array}$$

具体推导过程在论文Large System Performance of Linear Multiuser Receivers in Multipath Fading Channels中

其中步骤(a)的推导来自于：