Entrance examination Mines-Ponts MP 25: Mathématiques 1.
A probability problem set with many technicalities, using Rademacher random variables: \[ \begin{alignat*}{1} X(\Omega)&=\{-1,1\}\\ \mathrm{P}(X=1)&=\mathrm{P}(X=-1)=\frac{1}{2} \end{alignat*} \]
The main result was Khintchine's inequalities, a kind of equivalence of norms, which can be stated as follows: \[ \begin{alignat*}{1} \forall p\geqslant 1, \exists (\alpha_p,\beta_p)\in\mathbb{R}^{+*},\forall n\in\mathbb{N}, \forall (c_1,c_2,\dots,c_n)\in\mathbb{R}^n,\\ \alpha_p \left(\mathrm{E}\left(\left(\sum_{i=1}^nc_iX_i\right)^2\right) \right)^{\frac{1}{2}} \leqslant \left(\mathrm{E}\left(\abs{\sum_{i=1}^nc_iX_i}^p\right) \right)^{\frac{1}{p}} &\leqslant \beta_p \left(\mathrm{E}\left(\left(\sum_{i=1}^nc_iX_i\right)^2\right) \right)^{\frac{1}{2}} \end{alignat*} \]
Solution of problem set Maths 1 MP given on April 22nd for the Mines-Ponts entrance exam 2025 :
Find the problem statement here: