$N$ being a non-zero natural integer, $\left( X _ { j } ^ { i } \right) _ { 1 \leqslant i \leqslant N , 1 \leqslant j \leqslant n }$ is a family of $n \times N$ mutually independent real random variables with the same distribution $\mathcal { R }$. For every $i \in \llbracket 1 , N \rrbracket$, we set $X ^ { i } = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } ^ { i } , \ldots , X _ { n } ^ { i } \right) ^ { \top }$. Deduce from the previous questions that $$\mathbb { P } \left( \bigcup _ { 1 \leqslant i < j \leqslant N } \left| \left\langle X ^ { i } \mid X ^ { j } \right\rangle \right| \geqslant \varepsilon \right) \leqslant N ( N - 1 ) \exp \left( - \frac { \varepsilon ^ { 2 } n } { 2 } \right) .$$
$N$ being a non-zero natural integer, $\left( X _ { j } ^ { i } \right) _ { 1 \leqslant i \leqslant N , 1 \leqslant j \leqslant n }$ is a family of $n \times N$ mutually independent real random variables with the same distribution $\mathcal { R }$. For every $i \in \llbracket 1 , N \rrbracket$, we set $X ^ { i } = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } ^ { i } , \ldots , X _ { n } ^ { i } \right) ^ { \top }$.
Deduce from the previous questions that
$$\mathbb { P } \left( \bigcup _ { 1 \leqslant i < j \leqslant N } \left| \left\langle X ^ { i } \mid X ^ { j } \right\rangle \right| \geqslant \varepsilon \right) \leqslant N ( N - 1 ) \exp \left( - \frac { \varepsilon ^ { 2 } n } { 2 } \right) .$$