Let $\phi : \mathbb{R} \rightarrow \mathbb{R}$ be given by $$\phi(t) = \frac{1}{2} + \frac{e^{t^2}}{4e^{t^2} - 2}$$ Prove that $\phi$ is a characteristic function.

My attempt:

I know that there are certain criteria that say that a function is a characteristic function of distribution, eg the Bochner criterion, but I do not know how to prove that a function is positively defined. I also have the Polya criterion, but here the condition that $\phi (\infty) = 0$ is not satisfied. Could You give me some hints?

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    Low-tech approach: Expanding the geometric series, one gets $$\varphi(t)=\frac12+\frac14\frac1{1-\frac12e^{-t^2}}=\frac34+\sum_{n\geqslant1}\frac1{2^{n+2}}e^{-nt^2}$$ Now, use (and/or prove) the result that if every $\varphi_n$ is a characteristic function and if $(p_n)$ are nonnegative with $\sum p_n=1$ then $$\varphi=\sum_{n\geqslant0}p_n\varphi_n$$ is a characteristic function as well. – Did Dec 6 at 15:24
  • @Did That approach needs slightly more care in this case because there are infinitely many of the $\varphi_n$, but it works. – J.G. Dec 6 at 15:25
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    @J.G. I do not know what you mean by "slightly more care" ("more" than what?) but indeed the result I mentioned holds, for infinitely many nonzero coefficients $p_n$ just like it holds for finitely many. – Did Dec 6 at 15:28
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    @Did It's a trivial result when we take a linear combination of any finite number of cfs by induction, but the extension to infinitely many requires discussion of limits. In this case I'd take the pointwise limit of a sequence of pdfs from Fourier inversion, then Fourier-transform back to $\varphi$. – J.G. Dec 6 at 15:35
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    "Discussion of limits" Not necessarily, if one builds explicitely a random variable whose CF is $\sum\limits_{n=0}^\infty p_n\varphi_n$. – Did Dec 6 at 16:06

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