Define the class of probability density functions $\mathcal{C}$: $\,p \in \mathcal{C}$ iff $p(x)=p(|x|)$, and $\log p(\!\sqrt{x})$ is convex on $[0,\infty)$.
Conjecture: if $p,q \in \mathcal{C}$, then $p * q \in \mathcal{C}$ (where $*$ denotes convolution).
This would be similar to the closure of the class of log concave densities (P$\mathrm{\acute{o}}$lya frequency functions $\mathrm{PF}_2$) under convolution, but applying to heavy-tailed densities.
Sums of Gaussian random variables are Gaussian, and sums of Gaussian Scale Mixtures (GSMs) are GSMs, specifically, it is easily shown that, $$ \xi\hspace{1pt} Z_1 + \gamma\hspace{1pt} Z_2 \overset{d}{=} (\xi^2+\gamma^2)^{\frac{1}{2}} Z $$ where $Z,Z_1,Z_2$ are standard Normal, and $\xi,\gamma$ are non-negative random variables, all independent. So the class of GSMs, denoted $\mathcal{G}_S$, is closed under convolution.
Gaussian Scale Mixtures, by the Bernstein-Widder Theorem, have densities $p(x)$ such that $p(\!\sqrt{x})$ is completely monotonic, and thus $\log p(\!\sqrt{x})$ is convex on $(0,\infty)$. So $\mathcal{G}_S \subset \mathcal{C}$.
Also, using the defining limit of the exponential function, GSMs are the limit as $n\to \infty$ of the properly normalized $n$-times monotonic in $\sqrt{x}$ classes, denoted $\mathcal{M}_n$, where $p\in\mathcal{M}_n$ iff, $$ p(x) = \int_0^{\infty} (1-t\hspace{1pt}x^2)_+^{n-1} \hspace{1pt} d\mu(t) $$ with $\mu$ non-decreasing and bounded below. It can be shown that the classes $\mathcal{M}_n$ are closed under convolution. And it can be shown that $\mathcal{C} \subset \mathcal{M}_2$.
So $\mathcal{G}_S = \mathcal{M}_{\infty} \subset \mathcal{C} \subset \mathcal{M}_2$, and $\mathcal{G}_S$ and $\mathcal{M}_2$ are closed under convolution.
The modern proof that log concave densities are closed under convolution used the fact that for log concave densities $p(x-t)$ is a totally positive kernel of second order ($\mathrm{TP}_2$), so $\int p(x-t) q(t-y)dt=r(x-y)$ is $\mathrm{TP}_2$. The densities in $\mathcal{C}$ are not sign regular on $(-\infty,\infty)$ so this approach doesn't seem to apply.
It seems that $\mathcal{C}$ cannot be represented as a scale mixture or convolution of simpler functions.
Convex (or convex in $\!\sqrt{x}$) functions can be written as integrals of Heaviside type functions. So $p(x) = \exp(\int f(tx)d\mu(t))$. So $p$ can be written as the limit of a product of functions (that are not densities). But the convolution of a product of functions does not simplify to the product of convolutions e.g.
Can anyone think of a representation of the densities in $\mathcal{C}$ that might me useful?
Can anyone come up with a counterexample (or proof) that $\mathcal{C}$ is closed under convolution.
