I'd like to know how one would go about showing that the following function, $f$, that is almost everywhere positive exists:
$g(x_1,\cdots,x_n)=\int_{-\infty}^{\infty} \prod _{j=1}^nf(u-x_j)du$ where $g:\mathbb{R}^n:\rightarrow \mathbb{R}$ satisfies:
(1) $g(x_1,\cdots,x_n)$ is the derivative is the nth order partial derivative of $\frac{\partial \ln(G(e^{x_1},\cdots,e^{x_n})}{\partial x_1x_2...x_n}$; where $G(e^{x_1},\cdots,e^{x_n})$ is symmetric, homogenous of degree 1, $G(0)=0$, $\lim G(y)\rightarrow \infty$ as $y \rightarrow \infty$, $G(y)>0$.
(2) $g\ge0$
If this isn't possible, can you suggest of ways to add restrictions so that such an $f$ exists? I know this is asking a lot, but I was wondering if someone would be willing to give some direction.
Thanks so much in advance!!!!!