Characteristic function of a gaussian law

Let $g(x)=e^{-x^2\mathrm{/}2}\mathrm{/}\sqrt{2\pi }$ and let $Z$ have its law on the canonical real probability space. Normalization is supplied by the Gaussian-density lemma. Exponential differentiation and the chain rule give $g^{\mathrm{\prime }}(x)=-xg(x)$. FTC, reflection substitution and the bridge yield ${\int }_{-R}^R\mathrm{\mid }x\mathrm{\mid }g(x)dx=\frac{2}{\sqrt{2\pi }}(1-e^{-R^2\mathrm{/}2})(R>0)\mathrm{.}$ Monotone convergence over positive integer R gives $\mathbb{E}\mathrm{\mid }Z\mathrm{\mid }=2\mathrm{/}\sqrt{2\pi }<\mathrm{\infty }$ by density integration. Apply the moments lemma at order one: ${\phi }^{\mathrm{\prime }}(t)=\int ix{e}^{itx}g(x)dx$, converting real positive/negative parts of the density integral separately.

On $[-R,R]$ both $g$ and the real and imaginary parts of $e^{itx}$ are continuously differentiable. Integration by parts, applied componentwise, gives ${\int }_{-R}^{R}ix{e}^{itx}g(x)dx=-i[e^{itx}g(x){]}_{-R}^R-t{\int }_{-R}^R{e}^{itx}g(x)dx\mathrm{.}$ The boundary term has modulus at most $2g(R)\to 0$. The left integrand is dominated by the integrable $\mathrm{\mid }x\mathrm{\mid }g(x)$, and the last integral by $g(x)$; DCT along integer R therefore gives ${\phi }^{\mathrm{\prime }}(t)=-t\phi (t)$ for every real t. No imaginary displacement of an integration contour is involved.

The real and imaginary components of $H(t)=e^{t^2\mathrm{/}2}\phi (t)$ are differentiable. The product and chain rules and step 2.1 give $H^{\mathrm{\prime }}(t)=e^{t^2\mathrm{/}2}(t\phi (t)+{\phi }^{\mathrm{\prime }}(t))=0$. The zero-derivative theorem applied to each component on $\mathbb{R}$ makes H constant. Its value at zero is $\phi (0)=\int g=1$, so $\phi (t)=e^{-t^2\mathrm{/}2}$. Finally affine scaling gives ${\phi }_{m+\sigma Z}(t)=e^{imt}{\phi }_Z(\sigma t)=e^{imt-{\sigma }^2{t}^2\mathrm{/}2}$. If $\sigma =0$, the random variable is constantly m and its transform is directly $e^{imt}$, agreeing with the formula. The stated AC is inherited from normalization and the compact integration bridge (including their countable-choice prerequisites).