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It is easy to spot a discrete integer valued random variable by looking at its characteristic function, as that is periodic with period $2 \pi$, i.e. for binomial distribution it is $\phi(t) = (1-p+p \, \mathrm{e}^{i t})^n$.

I recently came across Khintchine's result that $\phi(t) = \frac{\zeta(s + i t)}{\zeta(s)}$ is a characteristic function of a random variable for $s > 1$. After some fudging, I determined that it corresponds to $x_k = -\log(k)$, where $k$ follows Zipf distribution with parameter $s-1$. Indeed:

$$ \mathbb{E}( \mathrm{e}^{ -i t \log(k)} ) = \mathbb{E}( k^{-i t} ) = \sum_{k \ge 1} k^{-i t} \frac{k^{-s}}{\zeta(s)} = \frac{\zeta(s+i t)}{\zeta(s)} $$

This characteristic function, thus, also corresponds to a discrete random variable.

This brings up a question: Can one easily spot a discrete random variable from it's characteristic function ? Or is inverting the characteristic function the only way ? How does one go about doing the inversion ? Ordinary inverse Fourier transform would produce distributions, right ?

Thank you.

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2 Answers 2

up vote 2 down vote accepted

I'm not sure how to distinguish purely discrete random variables, but the formula $$\lim_{T\to\infty}{1\over 2T}\int^T_{-T}|\phi(t)|^2\,dt=\sum\mu(\{x\})^2$$ tells us whether or not the distribution $\mu$ has a discrete part.

Reference: Exercise 3.7 (page 98) of Probability: Theory and Examples (2nd edition) by Richard Durrett.

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The characteristic function is the Fourier transform of the probability density (with positive exponent, instead of the more usual convention of negative; but that's convention). This also holds if the variable is discrete, by using a distribution (sum of Dirac deltas) inside the Fourier integral, which in turn can be equivalently expressed as a Fourier sum. If the variable takes values in the integers, then we have the DTFT, and the transform $X(\omega)$ is a periodic function ($2 \pi$). To recover the original density, is nothing more nor less than computing the inverse transform. If you apply it directy to the full $X(w)$ we would get the sum of deltas; equivalently, we can integrate over a single period to get the values of the probability function. All this is explained in the linked page.

This, if you are insterested in getting the values of the probability function. If you just want to know if the C F corresponds to a discrete variable: for uniformly discretized support (eg, integers), the CF must be periodic. For arbitrarily discrete variables (less usual) the criterion is more complex - see Byron's answer.

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