Let $(\Omega, \mathcal{F}, P)$ be a probability space and $X:\Omega\to\mathbb{R}$ a random variable. How to prove that the characteristic function $\varphi_X(t) = E[e^{itX}] =\int_{\Omega}e^{itX(\omega)}dP(\omega)$, is non-negative definite?
I didn't know the concept of non-negative definiteness for complex functions, and the only definition I've found is the followig:
The function $f:\mathbb{R}\to\mathbb{C}$ is said to be non-negative definite when: $$\sum_{1\leq i, k\leq n}f(x_i-x_k)\xi_i\bar{\xi_k}\geq 0$$ for every choice of $n\in\mathbb{N}$, $x_1, \,...,\,x_n \in \mathbb{R}$ and $\xi_1, \,...,\,\xi_n \in \mathbb{C}$.
From this definition, I can't see how to prove the statement. Is there an equivalent definition which makes it easier? If not, how do we prove it?
$$\sum_{k,\ell}\xi_k\bar{\xi_{\ell}}\varphi_X(t_k-t_{\ell})=E\left(\sum_{k,\ell}\xi_k\bar{\xi_{\ell}}e^{i(t_k-t_{\ell})X}\right)=E\left(\left|\sum_k\xi_ke^{it_kX}\right|^2\right)\geqslant0$$