Prove that a characteristic function $\varphi$ is semi positiveness in the following form :
Taken $n$ real numbers $t_{1},\cdots,t_{n}$ and $n$ complex ones $z_1,\cdots z_n$ we have
$$\sum\limits_{h,k=1}^n \varphi(t_{h} - t_{k})z_{h}\overline{z_k}$$
I only managed to rewrite the sum as follows :
$$\sum\limits_{h=k}\varphi(0)z_h\overline{z_h} + 2 \sum\limits_{h>k} \text{Re}(\varphi(t_h-t_k)z_h\overline{z_k}) \overset{?}{\geq 0}$$
But how to proceed from here ?
Just use that $\lvert a\rvert^2 = a\overline{a} $
$$\sum\limits_{h,k = 1}^{n} y_k \overline{y_h}\varphi(t_k-t_h) = \sum\limits_{h,k=1}^n y_k\overline{y_h}\int e^{ix(t_k-t_h)} d\mu =$$
$$ = \int \sum\limits_{h,k=1}^n y_ke^{ixt_k}\overline{y_h e^{ix t_h}} d\mu = \int \lvert \sum\limits_{h,k=1}^ny_ke^{ixt_k}\rvert^2 d\mu \geq 0$$