Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Suppose in this question that $\sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } = 1$. Show that the integral $\int _ { 0 } ^ { + \infty } t ^ { 3 } \mathrm { e } ^ { - t ^ { 2 } / 2 } \mathrm {~d} t$ converges, then that $$\mathbf { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 4 } \right) \leq 8 \int _ { 0 } ^ { + \infty } t ^ { 3 } \mathrm { e } ^ { - t ^ { 2 } / 2 } \mathrm {~d} t$$
Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Suppose in this question that $\sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } = 1$. Show that the integral $\int _ { 0 } ^ { + \infty } t ^ { 3 } \mathrm { e } ^ { - t ^ { 2 } / 2 } \mathrm {~d} t$ converges, then that
$$\mathbf { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 4 } \right) \leq 8 \int _ { 0 } ^ { + \infty } t ^ { 3 } \mathrm { e } ^ { - t ^ { 2 } / 2 } \mathrm {~d} t$$