in (\[eq:dotec\]) and (\[eq:ddotec\]), we obtain (\[eq:ebminlp\]) and (\[eq:s0lp\]). $\square$

Next, we show that the equation (\[eq:stacond\]) that needs to be satisfied by ${{\alpha^{\ast}_{\text{opt}}}}$ has a unique solution for a certain class of fading distributions.

\[theo:stacond\] The equation $ {{\alpha^{\ast}_{\text{opt}}}}p_z({{\alpha^{\ast}_{\text{opt}}}})=P\{z>{{\alpha^{\ast}_{\text{opt}}}}\} $ has a unique solution when $z$ has a Gamma distribution with integer parameter $n$, i.e., when the probability density function of $z$ is given by $$\begin{gathered}
\label{eq:gammadensity0}
p_z(z) = \frac{\lambda^n}{\Gamma(n)} z^{n-1} e^{-\lambda z}\end{gathered}$$ where $n \ge 1$ is an integer, $\lambda > 0$, and $\Gamma$ is the Gamma function [@Grimmett].

*Proof*: See Appendix \[app:stacond\].

*Remark:* In the special case in which $n = \lambda = m$ and $m
\ge 1$ is an integer, the Gamma density (\[eq:gammadensity0\]) becomes $$p_z(z)=\frac{m^m z^{m-1}}{\Gamma(m)}e^{-mz}$$ which is the probability density function of $z = |h|^2$ in Nakagami-$m$ fading channels (with integer $m$)[@book]. Moreover, when $m = 1$, we have the Rayleigh fading channel in which $z$ has an exponential distribution, i.e., $p_{z}(z) = e^{-z}$. Therefore, the result of Theorem \[theo:stacond\] applies for these channels.

*Remark:* Theorem \[theo:lowpower\] shows that the $\frac{E_b}{N_0}\Big|_{{{\sf{R}}}= 0}$ for any $\theta \ge 0$ depends only on ${{\alpha^{\ast}_{\text{opt}}}}$. From Theorem \[theo:stacond\], we know that if $z$ has the Gamma density function 