Why a characteristic function should be positive-definite?

Question

Suppose $X$ is a $\mathbb{R}^d$-valued random variable. Suppose $(t_k)_{1 \leq k \leq n} \subset \mathbb{R}^d$ and $(z_k)_{1 \leq k \leq n} \subset \mathbb{C}$. Why should hold that if $\phi$ is the characteristic function of $X$, $\sum \phi (t_h-t_k)z_h\overline{z_k}$?

Answer 1

Note that $$\displaystyle \sum \mathbb{E}(e^{i(t_i -t_j,x)})\lambda_i \bar{\lambda}_j = \mathbb{E}|\sum \lambda_k e^{it_k x}|^2 \ge 0$$