Product of two characteristic functions

Question

Suppose that $\mathbf{X}$ is a $\mathbb{R}^k$-valued random vector and $\mathbf{Y}$ is a $\mathbb{R}^\ell$-valued random vector. $\mathbf{X}$ and $\mathbf{Y}$ are not independent. For any $\mathbf{t}\in\mathbb{R}^k$ and any $\mathbf{s}\in\mathbb{R}^\ell$, characteristic functions of $\mathbf{X}$ and $\mathbf{Y}$ are defined by $$ \phi_{\mathbf{X}}(\mathbf{t})=\mathbb{E}\bigl[e^{i\mathbf{t}'\mathbf{X}}\bigr], \qquad\text{and}\qquad \phi_{\mathbf{Y}}(\mathbf{s})=\mathbb{E}\bigl[e^{i\mathbf{s}'\mathbf{Y}}\bigr]. $$ My question if that: is $\phi_{\mathbf{X}}(\mathbf{t})\phi_{\mathbf{Y}}(\mathbf{s})$ a characteristic function of some $\mathbb{R}^{k+\ell}$-valued random vector?

Answer 1

Yes. There are random vectors $\mathbf X'$ and $\mathbf Y'$ that are independent and have the same distributions as $\mathbf X$ and $\mathbf Y$, respectively. The distribution of $(\mathbf X',\mathbf Y')$ is the product of the marginal distributions of $\mathbf X$ and $\mathbf Y$, and has the desired joint characteristic function.