There are some problem , which I want to speak about:
1) Consider $\phi_{X}(t)$ - characteristic function. Prove that $|\phi_{X}(t)|^{2}$ is characteristic too.
I have two ideas about it : first of all I thought about $\widehat{\phi(t)\cdot \phi(t)} = 2\pi \widehat{\phi}*\widehat{\phi} (t)$ and then get inverse f.t. But that's not show the necessary result. Also I thought about get $Y$ independent with $X$ and having the same distribution, then because of $|\phi(t)|^{2} = \phi(t)\bar{\phi(t)}$ and because $X$ and $Y$ have one distribution and independent we obtain that : $\phi_{X+Y}(t)= \phi_{X}(t) \cdot \phi_{Y}(t) = \phi^{2}_{X}(t)$. I guess it should be something like that.
2) Consider $c_{i} \ge 0$ and $\phi_{i}(t)$ is characteristic function for $1 \le i \le n$. We have that $\sum c_{i} = 1$. Prove that $\sum_{i} c_{i} \phi_{i}$ is characteristic too.
I've tried to use Fourier transform , but it's not so easy to find corresponding random variable. I know that's easy to prove with Khinchin's theorem, but I guess there is exist more beautiful proof.
Any hints ?
Part 1
Suppose that $X_1,\,X_2 \sim X$ are independent random variables with the same distribution, and consider the random variable $X_1 - X_2$. Using independence of the variables we have
\begin{align} \phi_{X_1 - X_2}(t) & = \mathbf E \left[ e^{it(X_1 - X_2)} \right] \\ & = \mathbf E \left[ e^{itX_1}\right] \mathbf E \left[ e^{-itX_2}\right] \\ & = \phi_X(t) \phi_X(-t) \\ & = |\phi_X(t)|^2. \end{align} The final line follows from the basic property of characteristic functions that $\phi_X(-t) = \overline{ \phi_X(t)}$.
Part 2
Suppose that $\phi_i(t)$ is the characteristic function of a random variable $X_i$, $1 \leq i \leq n$, and that the $X_i$ are independent.
Now let $Y \in \{1,\ldots, n\}$ be a random variable with $\mathbf P (Y = i) = c_i$, independent of the $X_i$. Consider the random variable $Z = X_{Y}$. Again independence, and the law of total probability gives
\begin{align} \phi_Z(t) & = \mathbf E \left[ e^{itX_Y} \right] \\ & =\sum_{i=1}^n c_i \mathbf E \left[ e^{it X_Y} \, \big| \, Y = i\right] \\ & = \sum_{i=1}^n c_i \mathbf E \left[ e^{it X_i} \right] \\ & = \sum_{i=1}^n c_i \phi_i(t). \end{align}