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Let's consider a function: $$\phi(t) = \frac{\cos(t)}{1 + t^4} \tag{1}.$$ How can I check whether $(1)$ is a characteristic function? I tried using Polya's criterion. Unfortunately it doesn't work here.

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1 Answer 1

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$\phi$ is clearly a smooth function. Calculating the fourth derivative shows

$$\phi^{(4)}(0)=-23 <0.$$

By the following well-known statement, this implies that $\phi$ is not a characteristic function.

Lemma: Let $f(t) = \mathbb{E}e^{itX}$, $t \in \mathbb{R}$, be a characteristic function which is $4$ times differentiable. Then the $4$th derivative $f^{(4)}$ satisfies $$f^{(4)}(0) = \mathbb{E}(X^4) \geq 0.$$

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