$\chi^2$ distributed independent variables with $ 2(K_1-k+1) $ degrees of freedom with $K_1=\max(2n_R,n_T)$. By following the same line of arguments as in [@OymanNBPaulraj], we arrive at $$\begin{aligned}
\nonumber
    R_{sA} \geq R_{sA}^{lb}=  & \frac{1}{2}\sum_{j=1}^{L_1}\log_2\left(1+\frac{\rho}{n_T}
    \exp\left(\sum_{p=1}^{K_1-j}\frac{1}{p}-\gamma\right)\right)\\
& \quad[\text{case }n_T \ge 2n_R] \nonumber\end{aligned}$$ In [@LiYeCapStackedAlamou], a similar (however, looser) lower bound was derived for this case in order to analyze the asymptotic performance (with respect to $\rho$) of stacked OSTBC.

For the case of $n_R<n_T<2n_R$ we have $$\begin{aligned}
R_{sA}&
=\mathbb{E}\left[\frac{1}{2}\log_2\det\left(\mathbf{I}+\frac{\rho}{n_T}(\mathbf{H}')^H\mathbf{H}'\right)\right]\nonumber
\\ & =\mathbb{E}\left[\frac{1}{2}\log_2\det\left(\mathbf{I}+\frac{\rho}{n_T}\left(\mathbf{H} \mathbf{H}^H+
 \mathbf{H}_e\mathbf{H}_e^H\right)\right)\right]\nonumber \\
& =
 \mathbb{E}\left[\frac{1}{2}\log_2\det\left(\frac{1}{2}\mathbf{I}+\frac{\rho}{n_T}\mathbf{H} \mathbf{H}^H+
 \frac{1}{2}\mathbf{I}+\frac{\rho}{n_T}\mathbf{H}_e\mathbf{H}_e^H\right)\right]. \nonumber\end{aligned}$$ Applying now Minkowski’s determinant inequality results in $$\begin{aligned}
R_{sA}\geq \frac{1}{2}\mathbb{E}\left[\log_2\det\left(\mathbf{I}+\frac{2\rho}{n_T}\mathbf{H}\mathbf{H}^H\right)\right]\end{aligned}$$ and finally $$\begin{aligned}
\nonumber
    R_{sA} \geq R_{sA}^{lb}= & \frac{1}{2}\sum_{j=1}^{L}\log_2\left(1+\frac{2\rho}{n_T}
    \exp\left(\sum_{p=1}^{K-j}\frac{1}{p}-\gamma\right)\right) \\
& \quad[\text{case } n_R< n_T < 2n_R]. \nonumber\end{aligned}$$

The lower bound results derived in this subsection are summarized in Table \[tab:LowBounds\] on the top of the next page.

  -----------------------------------------------------------------------------------------------------------------
          Case                                           Lower bound on $R_{sA}$
  -------------------- --------------------------------------------------------------------------------------------
     $n_T \leq n_R$                     $\frac{n_T}{2}\log_2\left(1+\frac{\rho}{\frac{n_T}{2}}\exp
                            \left(\frac{1}{n_T}\sum_{j=1}^{n_T}\sum_{p=1}^{n_R-j}\frac{1}{p}-\gamma\right)\right)$

   $n_R < n_T < 2n_R$                   $\frac{1}{2}\sum_{j=1}^{L}\log_2\left(1+\frac{2\rho}{n_T}
                                         \exp\left(\sum_{p=1}^{K-j}\frac{1}{p}-\gamma\right)\right)$

    $2n_R \leq n_T $                    $\frac{1}{2}\sum_{j=1}^{L_1}\log_2\left(1+\frac{\rho}{n_T}
                                        \exp\left(\sum_{p=1}^{K_1-j}\frac{1}{p}-\gamma\right)\right)$

      $4n_R < n_T$             $n_R\log_2\left(1+\frac{2\rho}{n_T}\exp\left(\frac{1}{2n_R}\sum_{j=1}^{2n_R}
                                      \sum_{p=1}^{\nicefrac{n_T}{2}-j}\frac{1}{p}-\gamma\right)\right)$
  -----------------------------------------------------------------------------------------------------------------

Note that for high SNR, most of the bounds have a slope equal to $\nicefrac{L_1}{2}$, which equals the slope of the upper bound . Only for the case $n_R < 