grandes-ecoles 2019 Q7

grandes-ecoles · France · centrale-maths2__mp Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions

Does the sequence of functions $(\varphi_n)_{n \geqslant 1}$ defined by $$\forall n \in \mathbb{N}^{\star}, \quad \varphi_n : \begin{aligned} \mathbb{R} &\rightarrow \mathbb{R} \\ t &\mapsto \mathbb{E}\left(\cos\left(t X_n\right)\right) \end{aligned}$$ converge uniformly on $\mathbb{R}$?

Does the sequence of functions $(\varphi_n)_{n \geqslant 1}$ defined by
$$\forall n \in \mathbb{N}^{\star}, \quad \varphi_n : \begin{aligned} \mathbb{R} &\rightarrow \mathbb{R} \\ t &\mapsto \mathbb{E}\left(\cos\left(t X_n\right)\right) \end{aligned}$$
converge uniformly on $\mathbb{R}$?

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