ribution. We then have $$\begin{aligned}
\label{eq:Epsilarge}
E_\Psi &\approx \sqrt{2}\left(\left(1-\beta\right){\mathrm{erf}}^{-1}\left(1-\frac{2}{\Psi}\right)+\beta\mathrm{erf}^{-1}\left(1-\frac{2}{e\Psi}\right)\right),\end{aligned}$$ where $\beta$ denotes the Euler’s constant. Substituting into completes the proof.

Proof of Corollary \[Cor:aveGlsngrO\] {#App:proofCor:aveGlsngrO}
=====================================

From the tail region approximation for the inverse error function, we note that $$\label{eq:appinverflargex}
  \mathrm{erf}^{-1}(x)=  \sqrt{-\ln\left(1-x^2\right)}  \quad \text{as} \quad x\rightarrow 1.$$ Based on and , as $\Psi\rightarrow\infty$, $G_{\bar{R}}(\Psi)$ is asymptotically equivalent to $$\begin{aligned}
G_{\bar{R}}(\Psi)&\sim 1+\frac{\sqrt{2{\sigma^2_{R_\psi}}}}{{\bar{R}_{\psi}}}\left( \left(1-\beta\right)\sqrt{-\ln(4)+\ln\left(\frac{\Psi^2}{\Psi-1}\right)}\right.\notag\\
&\left.
+\beta\sqrt{1-\ln(4)+\ln\left(\frac{\Psi^2}{\Psi-1/e}\right)}\right)\notag\\
&\sim \frac{\sqrt{2{\sigma^2_{R_\psi}}}}{{\bar{R}_{\psi}}}\sqrt{\ln(\Psi)}.\end{aligned}$$ This completes the proof.

Proof of Proposition \[Prop:2\] {#App:proofoutGa}
===============================

With the Gaussian approximated PDF of $R_{\psi}$, we have the approximated outage probabilities for the systems without and with reconfigurable antennas as $$\label{eq:outageGsin}
  {\mathbb{P}}(R_{\psi}<R)=F_{R_\psi}(R)$$ and $$\label{eq:outageGrec}
  {\mathbb{P}}(R_{{\widehat{\psi}}}<R)=\left(F_{R_\psi}(R)\right)^{\Psi},$$ respectively, where $F_{R_\psi}(x)$ is given in . Substituting and into and , respectively, we can obtain the approximated $R_{{\widehat{\psi}}}^{{\mathrm{out}}}$ and $R_{\psi}^{{\mathrm{out}}}$ as $$\label{eq:derRcout}
  R_{{\widehat{\psi}}}^{{\mathrm{out}}}\approx F_{R_\psi}^{-1}(\epsilon^{\frac{1}{\Psi}})={\bar{R}_{\psi}}-\sqrt{2{\sigma^2_{R_\psi}}}\mathrm{erf}^{-1}\left(1-2\epsilon^{\frac{1}{\Psi}}\right)$$ and $$\label{eq:derRout}
R_{\psi}^{{\mathrm{out}}}\approx F_{R_\psi}^{-1}(\epsilon)={\bar{R}_{\psi}