I am given this function $$\phi(t) = (1+|t|)e^{-|t|}$$ and asked to prove that it is a characteristic function for a certain random variable. Bochner's theorem says that the main condition a function $f:\mathbb{R}\to\mathbb{C}$ needs to satisfy for this is being positive semi-definite, which means that for any $t_1,...,t_n$, the $n\times n$ matrix where $a_{ij} = \phi(t_i-t_j)$ is PSD.
I have tried to bound the expression $z^{T}\mathbb{\Phi}z$, where $z\in\mathbb{C}^n$ from below somehow but failed. Any insight would be appreciated.