Second-order characteristic-function expansion

Taylor at zero through degree two for cosine and sine, with third derivatives bounded by one, gives $\mathrm{\mid }\cos u-1+u^2\mathrm{/}2\mathrm{\mid }\le \mathrm{\mid }u{\mathrm{\mid }}^3\mathrm{/}6$ and $\mathrm{\mid }\sin u-u\mathrm{\mid }\le \mathrm{\mid }u{\mathrm{\mid }}^3\mathrm{/}6$. Euler form and the triangle inequality give $\mathrm{\mid }r(u)\mathrm{\mid }\le \mathrm{\mid }u{\mathrm{\mid }}^3\mathrm{/}3$. Taylor through degree one gives $\mathrm{\mid }\cos u-1\mathrm{\mid }\le u^2\mathrm{/}2$ and $\mathrm{\mid }\sin u-u\mathrm{\mid }\le u^2\mathrm{/}2$, hence $\mathrm{\mid }{e}^{iu}-1-iu\mathrm{\mid }\le u^2$. Adding $u^2\mathrm{/}2$ bounds $\mathrm{\mid }r(u)\mathrm{\mid }\le 3u^2\mathrm{/}2\le 4u^2$ for all real u. At u=0 all remainders vanish.

For $0<\mathrm{\mid }t\mathrm{\mid }\le 1\mathrm{/}n$, the first step gives $\mathrm{\mid }r(tX)\mathrm{\mid }\mathrm{/}{t}^2\le X^2\min (\mathrm{\mid }X\mathrm{\mid }\mathrm{/}(3n),4)$. This is a measurable nonnegative sequence tending pointwise to zero, bounded by the integrable $4X^2$. DCT therefore makes its expectations tend to zero. The bound is uniform over all such t, so $\mathbb{E}r(tX)=o(t^2)$ as a genuine two-sided real limit, without choosing a sequence of frequencies or requiring $\mathbb{E}\mathrm{\mid }X{\mathrm{\mid }}^3<\mathrm{\infty }$. Also $\mathrm{\mid }X\mathrm{\mid }\le 1+X^2$ gives integrability of the linear term.

By linearity, ${\phi }_X(t)=1+it\mathbb{E}X-t^2\mathbb{E}{X}^2\mathrm{/}2+\mathbb{E}r(tX)$. Insert the stated moments and the preceding remainder limit. If $\sigma =0$, the same majorants have zero integral, so the remainder vanishes and the formula still holds. At t=0 the defining expectation is one. Every limiting integrand was explicitly specified; no choice principle enters.