Approximation or bound on characteristic function difference

The question asks to bound the difference between a characteristic function and a Gaussian characteristic function (or another target), typically using Taylor expansion and moment bounds.

grandes-ecoles 2022 Q21 View

We are given a centered real random variable $Y$ such that $Y ^ { 4 }$ has finite expectation.
Conclude that for all real $\theta$,
$$\left| \Phi _ { Y } ( \theta ) - \exp \left( - \frac { \mathbf { E } \left( Y ^ { 2 } \right) \theta ^ { 2 } } { 2 } \right) \right| \leq \frac { | \theta | ^ { 3 } } { 3 } \left( \mathbf { E } \left( Y ^ { 4 } \right) \right) ^ { 3 / 4 } + \frac { \theta ^ { 4 } } { 8 } \mathbf { E } \left( Y ^ { 4 } \right)$$