Let $\left( X _ { i } \right) _ { i \in [ 1 , n ] }$ be a sequence of independent random variables all following a Rademacher distribution. Deduce that: for all $t \geq 0$, for all $x \geq 0$ and for all $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$, $$\mathbf { P } \left( \exp \left( x \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| \right) > \mathrm { e } ^ { t x } \right) \leq 2 \mathrm { e } ^ { - t x } \exp \left( \frac { x ^ { 2 } \sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } } { 2 } \right) .$$ You may use Markov's inequality.
Let $\left( X _ { i } \right) _ { i \in [ 1 , n ] }$ be a sequence of independent random variables all following a Rademacher distribution. Deduce that: for all $t \geq 0$, for all $x \geq 0$ and for all $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$,
$$\mathbf { P } \left( \exp \left( x \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| \right) > \mathrm { e } ^ { t x } \right) \leq 2 \mathrm { e } ^ { - t x } \exp \left( \frac { x ^ { 2 } \sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } } { 2 } \right) .$$
You may use Markov's inequality.