grandes-ecoles 2019 Q27

grandes-ecoles · France · centrale-maths2__mp Continuous Uniform Random Variables Convergence in Distribution or Probability

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$, $F_n(x) = \mathbb{P}(Y_n \leqslant x)$ and $G_n(x) = \mathbb{P}(Y_n < x)$.
Generalize the results obtained in Q26 for all $x \in [0,1]$.

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$, $F_n(x) = \mathbb{P}(Y_n \leqslant x)$ and $G_n(x) = \mathbb{P}(Y_n < x)$.

Generalize the results obtained in Q26 for all $x \in [0,1]$.

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