Let \(\mathbf{X}\) be an \(\mathbb{R}^{d}\)-valued random variable that is symmetric about the origin (that is, \(\boldsymbol{X}\) and \((-\boldsymbol{X})\) are identically distributed). Denote by \(\mu\) is its distribution and \(\psi(\boldsymbol{t})=\) \(\mathbb{E} \mathrm{e}^{i \boldsymbol{t}^{\top} \boldsymbol{X}}=\int \mathrm{e}^{i t^{\top} \boldsymbol{x}} \mu(d \boldsymbol{x})\) for \(\boldsymbol{t} \in \mathbb{R}^{d}\) is its characteristic function. Verify that \(\kappa\left(\boldsymbol{x}, \boldsymbol{x}^{\prime}\right)=\psi(\boldsymbol{x}\) \(-\boldsymbol{x}^{\prime}\) ) is a real-valued positive semidefinite function.