$N$ 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 $X ^ { i } = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } ^ { i } , \ldots , X _ { n } ^ { i } \right) ^ { \top }$. We assume that $n \geqslant 4 \frac { \ln N } { \varepsilon ^ { 2 } }$. Prove that $$\mathbb { P } \left( \bigcup _ { 1 \leqslant i < j \leqslant N } \left| \left\langle X ^ { i } \mid X ^ { j } \right\rangle \right| \geqslant \varepsilon \right) < 1 .$$
$N$ 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 $X ^ { i } = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } ^ { i } , \ldots , X _ { n } ^ { i } \right) ^ { \top }$.
We assume that $n \geqslant 4 \frac { \ln N } { \varepsilon ^ { 2 } }$. Prove that
$$\mathbb { P } \left( \bigcup _ { 1 \leqslant i < j \leqslant N } \left| \left\langle X ^ { i } \mid X ^ { j } \right\rangle \right| \geqslant \varepsilon \right) < 1 .$$