\Rightarrow \det \mathbf{S}< \det \mathbf{R}\big)$ we obtain $\det \mathbf{P}=(\det\mathbf{A})(\det[\mathbf{D}-\mathbf{BA}^{-1}\mathbf{B}^H])
   < (\det \mathbf{A})(\det \mathbf{D})$, if $\mathbf{B}$ has full rank.

From $\det\mathbf{B}=|\alpha_1|^2+|\alpha_2|^2$ it follows, that apart from the set of events $\{\alpha_1=\alpha_2=0\}$ of measure zero, $\mathbf{B}$ has full rank, thus the upper bound for $I_{sA}$ is strict with probability one.

### Case of arbitrary $n_R$

The available portion of the mutual information achievable with $n_R\geq 1$ for the stacked Alamouti scheme is $$\label{eq:exactMutInfoStackAlam}
I_{sA} = \frac{1}{2}\log_2\det\left(\mathbf{I}_{n_T} +\frac{\rho}{n_T}(\mathbf{H}')^H\mathbf{H}'\right)\;.$$ Following the derivation above for arbitrary $n_R$ results in $$\label{eq:UbbBoundStAl}
    I_{sA}\leq I_{sA}^{ub}=\frac{L_1}{2}\log_2 \left(1+\frac{2\rho}{n_TL_1}||\mathbf{H}||^2\right),$$ where $L_1=\min(n_T,2n_R)$. By averaging (\[eq:UbbBoundStAl\]) over all channel realizations, an upper bound on the average rate $R_{sA}^{ub}\geq \mathbb{E}[I_{sA}]$ of the stacked Alamouti scheme similar to  may be obtained $$\begin{aligned}
\label{eq:UppBoundExactNRBel}
    R_{sA} \leq R_{sA}^{ub} =\mathbb{E}\left[I_{ub}\right] = & \frac{L_1}{2\ln(2)}\sum\limits_{k=0}^{n_Tn_R-1}
    \left(\frac{n_TL_1}{2\rho}\right)^{n_Tn_R-k-1}\\
    & e^{\frac{n_TL_1}{2\rho}} \Gamma\left(1-(n_Tn_R-k),\frac{n_TL_1}{2\rho}\right)\nonumber,\end{aligned}$$ which can be approximated using $\log_2(1+x)\approx \log_2(x)$ for $x\gg 1$ by $$\nonumber
    R_{sA}^{ub}\approx
    \frac{L_1}{2}\log_2\left(\frac{2\rho}{n_TL_1}\right)+\frac{L_1}{2\ln(2)}\left(\sum\limits_{p=1}^{n_Tn_R-1}\frac{1}{p}-\gamma\right).$$ Note that the approximation gets better for higher SNR and may be inaccurate for low SNR. Further note that, for high SNR, the slope of the upper bound  and its approximation is equal to $\nicefrac{L_1}{2}$.

Lower bounds on the ergodic capacity and the average rate of stacked OSTBC {#sec:MutInfoLB}
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Similarly to the last subsection, here we 