is a convex function in $x$ for $c>0$ and by applying Jensen’s inequality it holds that $\mathrm{E}\left[\log_2(1+ce^x)\right]\geq \log_2(1+c\exp(\mathrm{E}\left[x\right]))$, we have $$\begin{aligned}
R_{sA}& \geq
\frac{n_T}{2}\log_2\left(1+\rho\exp\mathbb{E}\left[\ln\left(2\det\left(\frac{1}{n_T}\mathbf{H}^H\mathbf{H}\right)^{\frac{1}{n_T}}
\right)\right]\right) \nonumber\\
& = \frac{n_T}{2}\log_2\left(1+\rho
2\exp\frac{1}{n_T}\mathbb{E}\left[\ln\left(\det\left(\frac{1}{n_T}\mathbf{H}^H\mathbf{H}\right)
\right)\right]\right).\nonumber
 \end{aligned}$$

From [@OymanNBPaulraj; @Goodman], we know that $$\nonumber
\mathbb{E}\left[\ln\left(\det\left(\frac{1}{n_T}\mathbf{H}^H\mathbf{H}\right)
\right)\right]=\sum_{j=1}^{n_T}\mathbb{E}\left[\ln X_j\right]-n_T\ln n_T,$$ where the $X_j$ are independent, $\chi^2$ distributed independent variables with $ 2(n_R-j+1) $ degrees of freedom. Using this yields $$\begin{aligned}
\nonumber
R_{sA} & \geq \frac{n_T}{2}\log_2\left(1+\frac{\rho}{\frac{n_T}{2}}
\exp\left(\frac{1}{n_T}\sum_{j=1}^{n_T}\mathbb{E}\left[\ln X_j\right]\right)\right).
 \end{aligned}$$ With $$\nonumber
\mathbb{E}\left[\ln X_j\right]=\psi(n_R-j+1),$$ where $\psi(\cdot)$ is the digamma function, which may be rewritten for integer arguments as follows $$\nonumber
\psi(x)=-\gamma+\sum_{p=1}^{x-1}\frac{1}{p}.$$ Using this results in the following lower bound for the average rate of the stacked scheme. $$\begin{aligned}
\nonumber
    R_{sA}\geq & \frac{n_T}{2}\log_2\left(1+\frac{\rho}{\frac{n_T}{2}}\exp
    \left(\frac{1}{n_T}\sum_{j=1}^{n_T}\sum_{p=1}^{n_R-j}\frac{1}{p}-\gamma\right)\right)\\
& \quad[\text{case }n_T \leq n_R]. \nonumber\end{aligned}$$

Similar steps can be pursued for $n_T\geq 4 n_R$ resulting in the following lower bound $$\begin{aligned}
\nonumber
    R_{sA}\geq &
    n_R\log_2\left(1+\frac{2\rho}{n_T}\exp\left(\frac{1}{2n_R}\sum_{j=1}^{2n_R}
    \sum_{p=1}^{\nicefrac{n_T}{2}-j}\frac{1}{p}-\gamma\right)\right)\\
& \quad[\text{case }n_T > 4n_R] \nonumber\end{aligned}$$

For the case of $n_T\geq 2n_R$ we rewrite as $$\begin{aligned}
\label{eq:BlockStrukSOSTBCRate}
I_{sA} =\frac{1}{2}\log_2\det\left(\mathbf{I}_{2n_R} +\frac{\rho}{n_T}\left[%
\begin{array}{cc}
  \mathbf{H}\mathbf{H}^H & \mathbf{H}\mathbf{H}_e^H \\
  \mathbf{H}_e\mathbf{H}^H & \mathbf{H}_e\mathbf{H}_e^H \\
\end{array}%
\right]\right).\end{aligned}$$ Since $\mathbb{E}\left[\mathbf{H}\mathbf{H}_e^H\right]=\mathbf{0}$ from proposition \[prop:RelHandHe\], we may proceed as in [@FoschiniGans98] to arrive at a lower bound given as $$\begin{aligned}
\nonumber
I_{sA}\geq \frac{1}{2}\sum_{k=1}^{L_1}\log_2\left(1+\frac{\rho}{n_T}X_{k}\right),\end{aligned}$$ where $X_{k}$ are again independent, 