grandes-ecoles 2017 QII.A.3

grandes-ecoles · France · centrale-maths1__psi Discrete Random Variables Monotonicity and Convergence of Sequences Defined via Expectations

Let $a$ be a real number. We suppose that $P(X \geqslant a) > 0$. Show that the sequence $\left(\frac{\ln\left(P\left(S_{n} \geqslant na\right)\right)}{n}\right)_{n \geqslant 1}$ is well-defined and admits a limit $\gamma_{a}$ that is negative or zero, satisfying $$\forall n \in \mathbb{N}^{*}, \quad P\left(S_{n} \geqslant na\right) \leqslant \mathrm{e}^{n\gamma_{a}}$$

Let $a$ be a real number. We suppose that $P(X \geqslant a) > 0$. Show that the sequence $\left(\frac{\ln\left(P\left(S_{n} \geqslant na\right)\right)}{n}\right)_{n \geqslant 1}$ is well-defined and admits a limit $\gamma_{a}$ that is negative or zero, satisfying
$$\forall n \in \mathbb{N}^{*}, \quad P\left(S_{n} \geqslant na\right) \leqslant \mathrm{e}^{n\gamma_{a}}$$

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QI.C.1 QI.C.2 QI.C.3 QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QII.C.1 QII.C.2 QII.C.3