Characteristic functions under affine maps and independent sums

The addition law gives $e^{it(aX+b)}=e^{itb}{e}^{i(at)X}$. Both random exponentials are bounded and integrable. Pulling out the constant $e^{itb}$ gives the affine identity, including $a=0$ and $b=0$.

For $n\ge 1$ put $c_j=\cos (t{X}_j)$ and $s_j=\sin (t{X}_j)$. Expand $e^{it{\sum }_j{X}_j}={\prod }_j(c_j+i{s}_j)={\sum }_{A\subseteq \{1,\dots ,n\}}{i}^{\mathrm{\mid }A\mathrm{\mid }}{\prod }_{j\in A}{s}_j{\prod }_{j\mathrm{\notin }A}{c}_j\mathrm{.}$ Each real factor is a bounded Borel function of its own coordinate, so the real factorization theorem applies to each of these finitely many products. Complex linearity then gives $\mathbb{E}{e}^{it{\sum }_j{X}_j}={\sum }_A{i}^{\mathrm{\mid }A\mathrm{\mid }}{\prod }_{j\in A}\mathbb{E}{s}_j{\prod }_{j\mathrm{\notin }A}\mathbb{E}{c}_j={\prod }_j(\mathbb{E}{c}_j+i\mathbb{E}{s}_j)={\prod }_j{\phi }_{X_j}(t)\mathrm{.}$

For $n=0$ the sum is zero, its exponential is one, and the empty product is one. For $n=1$ the asserted identity is the defining expectation itself.