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Together with coauthors I'm working on a paper where we use the following Proposition:

If a real-valued random variable $X$ has bounded support, then except in the trivial case that $X$ has all its mass in a single point, its moment generating function $$ M(z) = E(e^{zX})$$ has a zero in the complex plane.

Notice that the result is the same whether we are talking about the moment generating function, the characteristic function, the Laplace transform or the Fourier transform. Since the moments of $X$ grow at most exponentially, they are all entire functions and just rotations of each other.

It seems to us that the proposition must be both well-known and important, and we are baffled that we haven't been able to find it stated explicitly with a simple and self-contained proof.

Is there a statement and simple proof of the proposition in the literature?

The proposition is a consequence of the Hadamard factorization theorem: Since $M(z)$ is of order (at most) 1, it can be written as $e^{az+b}$ times a product involving its zeros. If there aren't any, we are left with $M(z) = e^{az}$, and $X$ must have all its mass at $a$.

But it can be proved with much easier complex analysis (see below), and this is why we're asking.

There is a lemma in William Feller's book An Introduction to Probability Theory and Its Applications vol II p 525, that implies our proposition and that seems to have been distilled out of Harald Cramér's proof of the decomposition theorem conjectured by Paul Lévy. It states that if $\exp(c\cdot X^2)$ has finite expectation for some $c>0$ (a weaker condition than boundedness), then either $X$ is normal or its characteristic function has a zero.

But both Cramér's original paper Über eine Eigenschaft der normalen Verteilungsfunktion and Feller's book simply refer to the Hadamard factorization theorem. Feller even says about the lemma that "Unfortunately its proof depends on analytic function theory and is therefore not quite in line with our treatment...".

There is a very reasonable (and useful to us) proof of the same lemma in the recent paper Three remarkable properties of the Normal distribution by Eric Benhamou, Beatrice Guez and Nicolas Paris, so we're certainly not complaining, just wondering if something even simpler has been published.

To establish our Proposition, some easy parts of the proof of the Hadamard theorem will do: If $M(z)$ has no zeros, we can introduce the function $$ K(z) = \int_0^z \frac{M'(t)}{M(t)}\,dt,$$ and we have $M(z) = e^{K(z)}$ throughout the complex plane (actually $K$ is known as the cumulant generating function). Assuming without loss of generality that $X$ is supported on $[-1,1]$, we obtain $$ \text{Re}(K(z)) \leq \left|z\right|, $$ which leads by the Borel-Carathéodory theorem to $$ \left| K(z) \right| \leq 4\left| z \right|.$$ This makes $K(z)/z$ a bounded entire function, and Liouville's theorem finishes the proof.

Spelling out the Borel-Carathéodory argument with a couple of more equations will reduce the whole thing to undergraduate complex analysis, and this is what we are thinking of doing.

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    $\begingroup$ Regarding this type of results, I would advise you to look in the book "Characteristic Functions" by Eugene Lukacs in which there is a thorough treatment of the interplay between these probability transforms and complex analysis. $\endgroup$ – user69642 May 28 '20 at 21:46
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It is the content of Theorem 7.2.3 page 202 of Eugene Lukacs book "Characteristic Function".

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