Using Table 1 below, give an approximate bound for the value of $m$ and the value of a real number $h > 0$ such that there exists a rank $n_0 \in \mathbb{N}^*$ satisfying for all $n \geqslant n_0$, $$P\left(S_n > 1{,}1 \times nm\right) \leqslant \mathrm{e}^{-nh}.$$
$x_i$
4,50
4,55
4,60
4,65
4,70
$\hat{\lambda}^*(x_i)$
$4{,}1 \times 10^{-12}$
$4{,}1 \times 10^{-12}$
$4{,}1 \times 10^{-12}$
$4{,}1 \times 10^{-12}$
$4{,}1 \times 10^{-12}$
$x_i$
4,75
4,80
4,85
4,90
4,95
$\hat{\lambda}^*(x_i)$
$5{,}1 \times 10^{-4}$
$5{,}5 \times 10^{-3}$
$1{,}1 \times 10^{-2}$
$1{,}6 \times 10^{-2}$
$2{,}1 \times 10^{-2}$
$x_i$
5,00
5,05
5,10
5,15
5,20
$\hat{\lambda}^*(x_i)$
$2{,}6 \times 10^{-2}$
$3{,}1 \times 10^{-2}$
$3{,}6 \times 10^{-2}$
$4{,}1 \times 10^{-2}$
$4{,}6 \times 10^{-2}$
$x_i$
5,25
5,30
5,35
5,40
5,45
$\hat{\lambda}^*(x_i)$
$5{,}1 \times 10^{-2}$
$5{,}6 \times 10^{-2}$
$6{,}1 \times 10^{-2}$
$6{,}6 \times 10^{-2}$
$7{,}1 \times 10^{-2}$
Using Table 1 below, give an approximate bound for the value of $m$ and the value of a real number $h > 0$ such that there exists a rank $n_0 \in \mathbb{N}^*$ satisfying for all $n \geqslant n_0$,
$$P\left(S_n > 1{,}1 \times nm\right) \leqslant \mathrm{e}^{-nh}.$$
\begin{tabular}{|l|l|l|l|l|l|}
\hline
$x_i$ & 4,50 & 4,55 & 4,60 & 4,65 & 4,70 \\
\hline
$\hat{\lambda}^*(x_i)$ & $4{,}1 \times 10^{-12}$ & $4{,}1 \times 10^{-12}$ & $4{,}1 \times 10^{-12}$ & $4{,}1 \times 10^{-12}$ & $4{,}1 \times 10^{-12}$ \\
\hline
$x_i$ & 4,75 & 4,80 & 4,85 & 4,90 & 4,95 \\
\hline
$\hat{\lambda}^*(x_i)$ & $5{,}1 \times 10^{-4}$ & $5{,}5 \times 10^{-3}$ & $1{,}1 \times 10^{-2}$ & $1{,}6 \times 10^{-2}$ & $2{,}1 \times 10^{-2}$ \\
\hline
$x_i$ & 5,00 & 5,05 & 5,10 & 5,15 & 5,20 \\
\hline
$\hat{\lambda}^*(x_i)$ & $2{,}6 \times 10^{-2}$ & $3{,}1 \times 10^{-2}$ & $3{,}6 \times 10^{-2}$ & $4{,}1 \times 10^{-2}$ & $4{,}6 \times 10^{-2}$ \\
\hline
$x_i$ & 5,25 & 5,30 & 5,35 & 5,40 & 5,45 \\
\hline
$\hat{\lambda}^*(x_i)$ & $5{,}1 \times 10^{-2}$ & $5{,}6 \times 10^{-2}$ & $6{,}1 \times 10^{-2}$ & $6{,}6 \times 10^{-2}$ & $7{,}1 \times 10^{-2}$ \\
\hline
\end{tabular}