I came across the following statement in a book:
If $\varphi(t)$ is a characteristic function, then $|\varphi(t)|$ is not necessarily a characteristic function.
Here's my argument:
By Bochner’s theorem, we can check $|\varphi(t)|$ satisfies the three conditions:
- continuous at $t=0$
- $|\varphi(0)|=1$
- positive definite \begin{align} \sum_{i,j}\xi_i\bar\xi_j|\varphi(t_i-t_j)| &=\sum_{i,j}\xi_i\bar\xi_j|\varphi(t_i)||\varphi(t_j)| \\ &=\left(\sum_i \xi_i|\varphi(t_i)|\right)\overline{\left(\sum_i \xi_i|\varphi(t_i)|\right)}\\ &=\left|\sum_i \xi_i|\varphi(t_i)|\right|^2\geq 0 \end{align} The equality holds only when $\xi_i=0,i=1,2,\ldots,n$.
What's the problem with this argument?