Representation of the Characteristic Function

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We introduce some criteria that a function $f$ is a characteristic function (ch.f.).

1. Bochner's Theorem
$f$ is a ch.f. $\iff$
(1) $f(0)=1$;
(2) $f$ is continuous at $t=0$;
(3) $f$ is positive definite (p.d., see Supp).

The p.d. property is nearly impossible to verify, thus we do not recommend that checking the conditions of Bochner's Theorem.  Practically, the following theorems might be useful to verify a characteristic function.

2. P$\dot{o}$lya's Theorem
If $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies
(1) $f(0)=1$;
(2) $f(t)\geq0$;
(3) $f(t)=f(-t)$ symmetric;
(4) $f$ is decreasing on $[0,\infty)$;
(5) $f$ is continuous on $[0,\infty)$;
(6) $f$ is convex on $[0,\infty)$,
then $f$ is a ch.f.

3. If $f_\alpha(t)=\exp{\{-|t|^\alpha\}}$, $0<\alpha\leq2$, then $f_\alpha(t)$ is a ch.f.

4. If $f$ is a ch.f., then so is $e^{\lambda(f-1)}$ for each $\lambda\geq0$.


[Supp] A function $f$ is positive definite (p.d.) iff for any finite set of real numbers $t_j$ and complex numbers $z_j$ (with conjugate complex $\bar{z}$), $1\leq j\;eq n$, we have $$\sum_{j=1}^n\sum_{k=1}^n f(t_j-t_k)z_j\bar{z}_k\geq0.$$