| \begin{array}{rl}&{\operatorname{Pr}_{\boldsymbol{\varepsilon},\boldsymbol{X},Y}\left\{\left|\boldsymbol{\it Pa}(X)\cdot\boldsymbol{v}\right|\geq\frac{1}{\sqrt{4D^{2}-3}}\right\}}\\&{=\sum_{x_{i_{3}},x_{i_{4}},\ldots}\operatorname{Pr}\{X_{i_{1}}=1,X_{i_{2}}=1,X_{i_{3}}=x_{i_{3}},\ldots\}\cdot\mathbb{I}\left\{|(1,1,x_{i_{3}},x_{i_{4}},\ldots)\cdot(v_{1},v_{2},v_{3},\ldots)|\geq\frac{1}{\sqrt{4D^{2}-3}}\right\}}\\&{\quad\quad+\sum_{x_{i_{3}},x_{i_{4}},\ldots}\operatorname{Pr}\{X_{i_{1}}=1,X_{i_{2}}=0,X_{i_{3}}=x_{i_{3}},\ldots\}\cdot\mathbb{I}\left\{|(1,0,x_{i_{3}},x_{i_{4}},\ldots)\cdot(v_{1},v_{2},v_{3},\ldots)|\geq\frac{1}{\sqrt{4D^{2}-3}}\right\}}\\&{\geq\sum_{x_{i_{3}},x_{i_{4}},\ldots}\zeta\operatorname{Pr}\{X_{i_{1}}=1,X_{i_{3}}=x_{i_{3}},X_{i_{4}}=x_{i_{4}}\ldots\}\cdot\mathbb{I}\left\{|(1,1,x_{i_{3}},x_{i_{4}},\ldots)\cdot(v_{1},v_{2},v_{3},\ldots)|\geq\frac{1}{\sqrt{4D^{2}-3}}\right\}}\\&{\quad\quad+\sum_{x_{i_{3}},x_{i_{4}},\ldots}\zeta\operatorname{Pr}\{X_{i_{1}}=1,X_{i_{3}}=x_{i_{3}},X_{i_{4}}=x_{i_{4}},\ldots\}\cdot\mathbb{I}\left\{|(1,0,x_{i_{3}},x_{i_{4}},\ldots)\cdot(v_{1},v_{2},v_{3},\ldots)|\geq\frac{1}{\sqrt{4D^{2}-3}}\right\}}\\&{=\zeta\cdot\sum_{x_{i_{3}},x_{i_{4}},\ldots}\operatorname{Pr}\{X_{i_{1}}=1,X_{i_{3}}=x_{i_{3}},X_{i_{4}}=x_{i_{4}},\ldots\}\left(\mathbb{I}\left\{|(1,1,x_{i_{3}},x_{i_{4}},\ldots)\cdot(v_{1},v_{2},v_{3},\ldots)|\geq\frac{1}{\sqrt{4D^{2}-3}}\right\}\right.}\\&{\quad\quad\left.+\mathbb{I}\left\{|(1,0,x_{i_{3}},x_{i_{4}},\ldots)\cdot(v_{1},v_{2},v_{3},\ldots)|\geq\frac{1}{\sqrt{4D^{2}-3}}\right\}\right)}\\&{\geq\zeta\sum_{x_{i_{3}},x_{i_{4}},\ldots}\operatorname{Pr}\{X_{i_{1}}=1,X_{i_{3}}=x_{i_{3}},X_{i_{4}}=x_{i_{4}},\ldots\}}\\&{=\zeta,}\end{array} |
