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Let $X$ be a random variable with continuous density $\rho(x)$. Assume that $X$ is symmetric and $\vert X\vert. Since it has a bounded support, all moments of $X$ are well-defined. Let $m_i$ denote the moment $i$ of $X$, i.e. $$ m_i = \int_{-L}^{L} x^i \rho(x) dx. $$ Let $\phi(t)=\int e^{jtx}\rho(x) dx$ be the characteristic function of $X$. Consider the Taylor expansion of $\phi(t)$, $$ \phi(t) = 1 - \frac{t^2}{2!}m_2 + \frac{t^4}{4!}m_4 - \frac{t^6}{6!}m_6+\cdots $$ I would like to lower bound $\phi(t)$ by $1 - \frac{t^2}{2!}m_2$. Particularly, can we say $\phi(t)\geq 1-\frac{t^2}{2!}m_2 $ for $t^2<\frac{2!}{m_2}$?

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