## 4 Analysis of DoF and User Scaling
### User Scaling Law
Then, the sum DoF for downlink transmission, denoted by \(\text{DoF}_{d}\), is lower-bounded by  
\[\text{DoF}_{d}\geq M\cdot P_{d}\cdot P_{u}. \tag{14}\]  
Now, let us characterize two probabilities \(P_{d}\) and \(P_{u}\). First, \(P_{d}\) can be rewritten as  
\[ \begin{array}{ll}P_{d}&=\lim_{SNR\rightarrow\infty}\Pr\left\{\text{SNR} \sum_{k=1,k\
eq i}^{M}\left|{\mathbf{h}_{\pi_{i}}^{(d)}}^{\dagger}\mathbf{v}_ {k}\right|^{2}\leq\epsilon_{1},\;\forall i\in\{1,\cdots,M\}\right\}\\ &=\lim_{SNR\rightarrow\infty}\Pr\left\{\sum_{k=1,k\
eq i}^{M}\left|{\mathbf{h }_{\pi_{i}}^{(d)}}^{\dagger}\mathbf{v}_{k}\right|^{2}\leq\epsilon_{1}\text{ SNR}^{-1},\;\forall i\in\{1,\cdots,M\}\right\},\end{array}  \tag{15}\]  
where \(\epsilon_{1}=\bar{\epsilon}_{1}\left(\beta_{\pi_{1}}^{(d)}\right)^{-1}\), which is independent of SNR. Here, the term \(\sum_{k=1,k\
eq i}^{M}\left|{\mathbf{h}_{\pi_{i}}^{(d)}}^{\dagger}\mathbf{v}_ {k}\right|^{2}\) corresponds to the downlink scheduling metric of selected MS \(\pi_{i}\) with no path-loss component and follows the chi-square distribution with \(2M\) degrees of freedom for \(i\in\{1,\cdots,M\}\) since the \(M\)-dimensional downlink channel vector \(\mathbf{h}_{\pi_{i}}^{(d)}\) is isotropically distributed. Note that the right-hand side of Equation (15) indicates the probability that there exist at least \(M\) MSs that fulfills the inequality \(\sum_{k=1,k\
eq i}^{M}\left|{\mathbf{h}_{\pi_{i}}^{(d)}}^{\dagger}\mathbf{v}_ {k}\right|^{2}\leq\epsilon_{1}\text{SNR}^{-1}\). Thus, by denoting \(F(x)\) by the cumulative density function (CDF) of a chi-square random variable with \(2M\) degrees of freedom, it follows that  
\[ \begin{array}{llll}P_{d}&=&1-\lim_{SNR\rightarrow\infty}\sum_{i=0}^{M-1} \binom{N}{i}F\left(\epsilon_{1}\text{SNR}^{-1}\right)^{i}\cdot\left(1-F\left( \epsilon_{1}\text{SNR}^{-1}\right)\right)^{N-i}\\ &=&1-\lim_{SNR\rightarrow\infty}\sum_{i=0}^{M-1}\frac{N!}{i!(N-i)!}\frac{F \!\left(\epsilon_{1}\text{SNR}^{-1}\right)^{i}\!\!\cdot\!\left(1-F\!\left( \epsilon_{1}\text{SNR}^{-1}\right)\right)^{N}}{\left(1-F\!\left(\epsilon_{1} \text{SNR}^{-1}\right)\right)^{i}}\\ &\stackrel{{(a)}}{{\geq}}&1-\lim_{SNR\rightarrow\infty}\sum_{i=0}^ {M-1}\frac{\left(N\cdot F\!\left(\epsilon_{1}\text{SNR}^{-1}\right)\right)^{i }\!\!\cdot\!\left(1-F\!\left(\epsilon_{1}\text{SNR}^{-1}\right)\right)^{N}}{ \left(1-F\!\left(\epsilon_{1}\text{SNR}^{-1}\right)\right)^{i}}\\ &\stackrel{{(b)}}{{\geq}}&1-\lim_{SNR\rightarrow\infty}\sum_{i=0}^ {M-1}\frac{\left(N\!C_{d,2}\text{SNR}^{-M}\right)^{i}\!\!\cdot\!\left(1-C_{d,1 }\text{SNR}^{-M}\right)^{N}}{\left(1-C_{d,2}\text{SNR}^{-M}\right)^{i}}, \end{array}  \tag{16}\]  
where  
\[C_{d,1} = \frac{e^{-1}2^{-M}}{M\cdot\Gamma(M)}\cdot\epsilon_{1}^{M} \tag{17}\]  
and  
\[C_{d,2} = \frac{2^{-(M-1)}}{M\cdot\Gamma(M)}\cdot\epsilon_{1}^{M}. \tag{18}\]  
Here, \(\Gamma(M)=\int_{0}^{\infty}t^{M-1}e^{-t}dt\) is the Gamma function; \((a)\) holds from the fact that \(\frac{N!}{i!(N-i)! }\leq N^{i}\); and \((b)\) holds from the fact that (see Lemma 1 in [20])  
\[\frac{e^{-1}2^{-M}}{M\cdot\Gamma(M)}\cdot x^{M}\leq F(x)\leq\frac{2^{-(M-1)}} {M\cdot\Gamma(M)}\cdot x^{M}. \tag{19}\]Next, let us turn to characterizing \(P_{u}\) as follows:  
\[P_{u} \geq \lim_{SNR\rightarrow\infty}\Pr\left\{\text{SNR}\sum_{j\in\mathcal{S}^ {(u)}}\left|h_{\pi j}\right|^{2}\leq\epsilon_{2},\;\forall i\in\{1,\cdots,M\}\right\}\] \[\geq \lim_{SNR\rightarrow\infty}\Pr\left\{\text{SNR}\sum_{j\in \mathcal{S}^{(u)}}\sum_{i=1}^{M}\left|h_{\pi j}\right|^{2}\leq\epsilon_{2}\right\}\] \[\geq \lim_{SNR\rightarrow\infty}\Pr\left\{\sum_{i=1}^{M}\left|h_{\pi j }\right|^{2}\leq\frac{\epsilon_{2}\text{SNR}^{-1}}{M},\;\forall j\in\mathcal{ S}^{(u)}\right\},\]  
where \(\epsilon_{2}=\tilde{\epsilon}_{2}\left(\max\{\beta_{\pi i\
ot\in_{1}}^{(M)}, \cdots,\beta_{\pi i\
ot\in_{M}}^{(M)}\}\right)^{-1}\), which is independent of SNR.