Let $X$ be a random variable with continuous density $\rho(x)$. Assume that $X$ is symmetric and $\vert X\vert
The second order approximation of the Taylor expansion of Characteristic functions:
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functional-analysis
taylor-expansion
characteristic-functions
1 Answers
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Assume only that $X$ is square-integrable with $\mathrm E(X)=0$ and $\mathrm E(X^2)=m_2$. Since $|\phi''(t)|\leqslant m_2$ for every $t$ and $\phi'(0)=0$, the mean value theorem for vector-valued functions shows that $|\phi'(t)|\leqslant m_2|t|$ for every $t$. Since $\phi(0)=1$, a second application of the mean value theorem yields $|\phi(t)-1|\leqslant\frac12m_2t^2$, hence $\phi(t)\geqslant1-\frac12m_2t^2$ for every $t$.