We are given a centered real random variable $Y$ such that $Y ^ { 4 }$ has finite expectation. Show, for all real $u$, the inequality $$\left| e ^ { i u } - 1 - i u + \frac { u ^ { 2 } } { 2 } \right| \leq \frac { | u | ^ { 3 } } { 6 }$$ Deduce that for all real $\theta$, $$\left| \Phi _ { Y } ( \theta ) - 1 + \frac { \mathbf { E } \left( Y ^ { 2 } \right) \theta ^ { 2 } } { 2 } \right| \leq \frac { | \theta | ^ { 3 } } { 3 } \left( \mathbf { E } \left( Y ^ { 4 } \right) \right) ^ { 3 / 4 }$$
We are given a centered real random variable $Y$ such that $Y ^ { 4 }$ has finite expectation.
Show, for all real $u$, the inequality
$$\left| e ^ { i u } - 1 - i u + \frac { u ^ { 2 } } { 2 } \right| \leq \frac { | u | ^ { 3 } } { 6 }$$
Deduce that for all real $\theta$,
$$\left| \Phi _ { Y } ( \theta ) - 1 + \frac { \mathbf { E } \left( Y ^ { 2 } \right) \theta ^ { 2 } } { 2 } \right| \leq \frac { | \theta | ^ { 3 } } { 3 } \left( \mathbf { E } \left( Y ^ { 4 } \right) \right) ^ { 3 / 4 }$$