Let $\phi$ is the characteristic function of a probability measure on $\mathbb{R}$, how can I prove $\sum_{i,j=1}^{n}{\phi(t_i-t_j)\bar{\phi}(t_i-t_j)\xi_i\bar{\xi_j}}\geq 0$ for all $n\in\mathbb{N}$, $\xi_1,\ldots,\xi_n\in\mathbb{C}$, and $t_1,\ldots,t_n\in\mathbb{R}$?
I already know the proof for $\phi$, but here we have two kernel functions. Can someone give me hint?
Let $X$ and $Y$ are two independent random variable whose common characteristic function is $\phi$. Then the characteristic function of $X-Y$ is $t\mapsto \phi(t)\overline{\phi(t)}$ hence the non-negativeness of $\sum_{i,j=1}^{n}{\phi(t_i-t_j)\bar{\phi}(t_i-t_j)\xi_i\bar{\xi_j}}\geqslant 0$ follows from the fact that a characteristic function is positive semi-definite.