Question: 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

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.

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