Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 27J / d / i / Solution

If $|\varphi_X(u)|=1$, then $e^{iuX}$ is almost surely constant. Indeed,
$$
\mathbb E\left[\left|e^{iuX}-\mathbb E[e^{iuX}]\right|^2\right]
=1-|\varphi_X(u)|^2=0.
$$
Hence $uX\in\theta+2\pi\mathbb Z$ almost surely for some real $\theta$.


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 Incoming links: Unit modulus of a characteristic function