I saw these properties in the wikipedia page but I was unable to prove them. I had an idea to construct a random variable with the desired characteristic function, yet I haven't managed to that so far. My only knowledge involving characteristic functions is that of the definition and some of its basic properties that you can derive from the definition. The properties are the following:
- Let $\phi_1,...,\phi_n$ be characteristic functions of some random variables $X_1,...,X_n$ and $p_i > 0$ with $p_1 + \ldots + p_n = 1$. Argue that $\phi = p_1 \phi_1 + \ldots + p_n \phi_n$ is also a characteristic function.
- If $\phi$ is a characteristic function and $\alpha$ is a real number, then $\mathrm{Re}(\phi)$, $|\phi|^2$, and $\phi(\alpha t)$ are also characteristic functions.
Thank you for your help!