grandes-ecoles 2020 Q36

grandes-ecoles · France · centrale-maths1__psi Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof

We define, for all $x \in \mathbb{R}$, $\lambda^*(x) = \sup_{t \geqslant 0}(tx - \lambda(t))$. Let $\varepsilon > 0$ and $n_0$ as in Q35. Using Markov's inequality applied to the random variable $e^{tS_n}$, show that for $a > 1$, $n \geqslant n_0$ and $t \geqslant 0$, $$P\left(S_n \geqslant nam\right) \leqslant \mathrm{e}^{-ntam} \mathrm{e}^{n(\lambda(t) + \varepsilon)}.$$

We define, for all $x \in \mathbb{R}$, $\lambda^*(x) = \sup_{t \geqslant 0}(tx - \lambda(t))$. Let $\varepsilon > 0$ and $n_0$ as in Q35. Using Markov's inequality applied to the random variable $e^{tS_n}$, show that for $a > 1$, $n \geqslant n_0$ and $t \geqslant 0$,
$$P\left(S_n \geqslant nam\right) \leqslant \mathrm{e}^{-ntam} \mathrm{e}^{n(\lambda(t) + \varepsilon)}.$$

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