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Let $f(t)=Ee^{itX}$ be a characteristic function.
Assume that $f(t)=1+\phi(t)+o(t^2)$, where $\phi(t)$ is odd, i.e. $\phi(t)=-\phi(-t)$. Prove that $f(t)=1$, $\forall t$

The assumption means that $E\cos(tX)=1+o(t^2)$ i.e. $E\frac{1-\cos(tX)}{t^2}\rightarrow 0$
I want to use the dominated convergence theorem to obtain $EX^2=0$ then $X=0$ a.s.
But it seems that this only works when $X$ has a finite monent of order 2.

Xiong Jiangnan
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1 Answers1

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$0\leq E\left(\frac{1-\cos(tX)}{t^2}1_{\{|X|\leq M\}}\right)\leq E\frac{1-\cos(tX)}{t^2}$
Let $t\rightarrow0$, we obtain by Dominated Convergence Theorem that $E(X^21_{\{|X|\leq M\}})=0$
Let $M\rightarrow\infty$, $E(X^2)=0$, $X=0$ a.s.

Xiong Jiangnan
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