8 problems

Let $X$ be a nonnegative random variable, and let ${\rm GGC}$ denote the class of generalized gamma convolutions. Bondesson's conjecture. If … then … for every $r\geq 1$. This is a…

Let $\alpha\in(0,1)$ and $\gamma>0$, and let $\widehat{\mathbf X}_{\alpha,\gamma}$ be the positive random variable whose density $\widehat h_{\alpha,\gamma}$ solves … for $x>0$. A…

Let $X$ be a positive random variable with a generalized gamma convolution law, and let $r>1$. A power transformation of order $r$ is the random variable $X^r$. Bond's power-transf…

Let $q>0$, and let $\varphi_q^{\pm}$ denote the Wiener-Hopf factors associated with the normal inverse Gaussian process, so that $\varphi_q^{\pm}(\pm z)$ are defined on the appropr…

Behme–Bondesson conjecture. The same conclusion remains true for all $k>0$:

Let ${\bf K}_\alpha$ be the random variable defined in the paper, with parameter $\alpha\in(0,1)$. A positive random variable has a generalized gamma convolution (GGC) law if its d…

Let $\operatorname{Beta}(\alpha_1,\alpha_2)$ denote a Beta random variable with parameters $\alpha_1,\alpha_2>0$, and let $s\ge1$. A positive random variable is generalized gamma c…

Let $\eta$ be a subordinator and let $\tau$ be an exponentially distributed random variable, independent of $\eta$. Consider the random variable $\eta_\tau$ and its generalized gam…