Bernstein-type inequality （BTI）

参见论文： Dual-Functional Artificial Noise (DFAN) Aided
Robust Covert Communications in Integrated
Sensing and Communications

理论

\boxed{} ​用于加框

Lemma 2. (BTI): For any $\mathbf{A}\in {\mathbb{C}}^{N\times N}$,$\mathbf{b}\in {\mathbb{C}}^{N\times 1}$,$c\in \mathbb{R}$,$\mathbf{x}\sim \mathcal{C}\mathcal{N}(0,\mathbf{I})$and$\rho \in [0,1]$,if there exist $x$ and $y$, such that

$$\begin{array}{rl}\mathrm{Tr}(\mathbf{A})-\sqrt{2\ln (\frac{1}{\rho })}x+\ln (\rho )y+c& \ge 0,\\ \sqrt{{\Vert \mathbf{A}\Vert }_F^2+2{\Vert \mathbf{b}\Vert }^2}& \le x,\\ y\mathbf{I}+\mathbf{A}⪰0,y& \ge 0,\end{array}$$
the following inequality holds true

Pr ⁡ ( x H A x + 2 R e { x H b } + c ≥ 0 ) ≥ 1 − ρ . \Pr(\mathbf{x}^H\mathbf{A}\mathbf{x}+2\mathrm{Re}\{\mathbf{x}^H\mathbf{b}\}+c\geq0)\geq1-\rho. Pr(xHAx+2Re{xHb}+c≥0)≥1−ρ.

应用例子：

已知：
$$\mathrm{Pr}({\mathbf{h}}_{\mathrm{w}}^H{\mathbf{S}}_1{\mathbf{h}}_{\mathrm{w}}\le 0)\ge 1-{\rho }_c,$$

$$\begin{array}{c}\text{Recall that }{\mathbf{h}}_{\mathrm{w}}={\hat{\mathbf{h}}}_{\mathrm{w}}+{\mathbf{\gamma }}_{\mathrm{w}}^{\frac{1}{2}}{\mathbf{e}}_{\mathrm{w}}.\text{ As per Lemma 2, (28) can}\\ \text{be equivalently transformed to the following inequalities}\end{array}$$

$$\begin{array}{rl}\mathrm{Tr}({\mathbf{A}}_{\mathrm{w}})-\sqrt{2\ln (\frac{1}{{\rho }_c})}x+\ln ({\rho }_c)y+c_{\mathrm{w}}& \ge 0,\\ \sqrt{{\Vert {\mathbf{A}}_{\mathrm{w}}\Vert }_F^2+2{\Vert {\mathbf{b}}_{\mathrm{w}}\Vert }^2}& \le x,\\ y\mathbf{I}+{\mathbf{A}}_{\mathrm{w}}& ⪰0,y\ge 0,\end{array}$$

$$\begin{array}{rl}& \mathrm{where}{\mathbf{A}}_{\mathrm{w}}={\gamma }_{\mathrm{w}}^{\frac{1}{2}}(-{\mathbf{S}}_1){\gamma }_{\mathrm{w}}^{\frac{1}{2}},c_{\mathrm{w}}={\hat{\mathbf{h}}}_{\mathrm{w}}^H(-{\mathbf{S}}_1){\hat{\mathbf{h}}}_{\mathrm{w}}\mathrm{and}{\mathbf{b}}_{\mathrm{w}}=\\ & {\gamma }_{\mathrm{w}}^{\frac{1}{2}}(-{\mathbf{S}}_1){\hat{\mathbf{h}}}_{\mathrm{w}}.\end{array}$$

注意其中只有$x$和$y$是辅助变量。