## Appendix B Detailed Longer Proofs of Theorems, Lemmas and Corollaries
### Proofs for Section 3
\(\mid\) denotes OR, e.g., 4 l 2 means 4 OR 2; \(\mid\) a b \(\mid\) c denotes a OR b OR c for any objects a, b, c.  
\(\mathbf{v}=\left\{a,b,\ldots,z\right\}\) denotes a column vector with elements \(a\), \(b\), \(\ldots\), \(z\). \(\mathbf{1}_{m}^{\forall}\) denotes the all-ones column \(m\)-vector: \(\mathbf{1}_{m}^{\forall}\equiv\underbrace{\left\{1,1,\ldots,1\right\}}_{m \text{ ones}}\). \(m\)-vector means a vector with \(m\) elements. \(\mathbf{1}_{C\times S}^{\forall}\) denotes an all-ones matrix of dimensions C \(\times\) S.  
\(\mathbf{M}_{m,z}(\mathbf{M}_{,z})\) designates the \(m\)-th row (\(n\)-th colon) of the matrix \(\mathbf{M}\), as per [13]. diagMat\(\{\mathbf{v}\}\) is a diagonal matrix with the elements of \(\mathbf{v}\) arrayed along its diagonal. \(\Sigma\mathbf{v}\) is the sum of elements of the \(m\)-vector \(\mathbf{v}\), and when \(\mathbf{v}\) is a column, \(\Sigma\mathbf{v}=\left(\mathbf{1}_{m}^{\forall}\right)^{T}\mathbf{v}\). \(\odot\) denotes the Shur (elementwise) product, \(\mathbf{v}\odot\mathbf{v}\), of compatible vectors. \(\mathbf{v}^{\odot 2}\) is the Shur-square: \(\mathbf{v}^{\odot 2}\equiv\mathbf{v}\odot\mathbf{v}\equiv\left\{v_{1}^{2},v_{2}^{2},\ldots,v_{m}^{2}\right\}\) for \(\mathbf{v}\equiv\left\{v_{1}^{1},v_{1}^{2},\ldots,v_{1}^{m}\right\}\). For any function \(f(\cdot):\mathbb{R}\rightarrow\mathbb{R}\), and vector \(\mathbf{v}\in\mathbb{R}\rightarrow\mathbb{R}^{n}\), \(f(\mathbf{v})\equiv\left\{f(v_{1}),f(v_{2}),...,f(v_{n})\right\}\) is defined elementwise, e.g., the sine of a vector is the vector of the sines of the elements of the vector. \(X:=Y\) means substitute Y for X; \(Z(X)\big{|}_{X:=Y}\) means \(Z(Y)\). \(\left\langle R\right\rangle\) denotes the (stochastic) Mean (expectation) of an RV, \(R\), and more compactly we also use an alternative superscript notation: \(R^{(\,)}\equiv\left\langle R\right\rangle\equiv\) mean of \(R\). The mapping  
\[\delta:\mathbb{R}\rightarrow\mathbb{R},\delta R\equiv R-\left\langle R\right\rangle\] (A97)  
is called the \"perturbation operator\". Any generic RV, \(R\), is decomposed as mean, \(\left\langle R\right\rangle\)+ perturbation, \(\delta R\): \(R=\left\langle R\right\rangle+\delta R\). The variance of \(R\) is denoted, \(\text{var}\left\{R\right\}\equiv\sigma_{R}^{2}\equiv\left\langle\left|\delta R \right|^{2}\right\rangle\) and we always have \(\left\langle\delta R\right\rangle=0\). Let \(\mathbf{R}=\left\{R_{1},R_{2}\ldots,R_{M}\right\}\) be a real-valued random vector. Its variance vector is defined as the vector of variances of its components: \(\sigma_{\mathbf{R}}^{2}\equiv\left\{\sigma_{R_{1}}^{2},\sigma_{R_{2}}^{2}, \ldots,\sigma_{R_{M}}^{2}\right\}\). A random vector, \(\mathbf{R}\), is called \"independent\" when its elements \(\left\{R_{1},R_{2}\ldots,R_{M}\right\}\) are independent RVs. For any deterministic \(M\)-vector, \(\mathbf{D}\),  
\[\sigma_{\mathbf{R}+\mathbf{D}}^{2}=\sigma_{\mathbf{R}}^{2}\] (A98)  
\(\text{MS}_{\mathbf{R}}\) denotes the Mean-Square (second-moment element-wise) of a random vector:  
\[\text{RMS}\left\{\mathbf{R}\right\}\equiv\sqrt{\left\langle\mathbf{R}^{2} \right\rangle}=\sqrt{\text{MS}_{\mathbf{R}}}\] (A100)is the Root-Mean-Square (RMS) of a vector, element-wise (take the Squares of the components, the (stochastic) Means and the (square) Roots). \(\overline{\mathbf{R}}\equiv\sum_{i=1}^{m}R_{i}\) denotes the (arithmetic) Average of the sequence \(\mathbf{R}\). \[\text{RAS}\{\mathbf{R}\}\equiv\sqrt{\overline{\mathbf{R}^{2}}}=\sqrt{\frac{1} {M}{\sum_{i=1}^{M}R_{i}^{2}}} \tag{101}\]  
is the Root-Average-Square (RMS) of a vector (Square the components, eval their (arithmetic) Mean and then (square) Root).