19 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 $\widehat h_{\alpha,\gamma}$ be the density on $(0,\infty)$ of the positive random variable $\widehat{\mathbf X}_{\alpha,\gamma}$, sati…

Rastegar and Roitershtein formulate a characterization theorem with a condition numbered (1.3), also numbered (3) in the preprint, and consider the range $n\ge3$. Rastegar–Roitersh…

Infinite divisibility conjecture. The random variable ${\hat T}_\theta$ is infinitely divisible for every

Let $\tau(\alpha,\rho)$ be the first hitting time of zero for a real strictly $\alpha$-stable process starting from one, where $\alpha\in(1,2]$ and $\rho$ is the positivity paramet…

The Le Roy-type Mittag-Leffler function is associated, when the relevant complete-monotonicity condition holds, with an underlying positive random variable. Logarithmic infinite-di…

Let $\mathcal{P}_{+}^{*m}$ denote the class of distributions formed from $m$-fold convolution products of distributions on the positive integers, and let $r_{\mathrm{ID}}(F)$ be th…

Let $X=(Y_{1},\ldots,Y_{m})$ be an i.i.d. sequence with $m\ge2$. The currently established multiplicative gap for dividing $X$ into i.i.d. pieces is $1+4.71\sqrt{\frac{\log m}{m}}$…

Let $\sigma$ be a probability measure on $(0,\infty)$, and let $g$ be a completely monotonic function. Steutel's conjecture. Every function of the form … is an infinitely divisible…

Let ${\rm \bf ID}(\rhd)$ denote the class of monotonically infinitely divisible probability measures on $\mathbb{R}$, and equip the space of probability measures with the weak topo…

Lin's conjecture. For every $t>0$, $\mathbf X_t$ is MD if and only if $\mathbf X^t$ is MD.

Let ${\bf X}_{\alpha,\rho}$ be the free stable random variable with stability parameter $\alpha>1$ and positivity parameter $\rho\in[1-1/\alpha,1/\alpha]$. A probability distributi…

Let $(X_1,X_2,X_3,X_4)$ be a jointly Gaussian vector, and consider the two-dimensional vector whose components are sums of two Gaussian squares, $(X_1^2+X_2^2,X_3^2+X_4^2)$. Infini…

Infinite divisibility conjecture. The distribution of

Let $T={\mathcal A}(\alpha,\rho,0)$, with the parameters understood as above. A probability distribution is infinitely divisible (ID) if, for every positive integer $n$, it is the…

Jurek–Sato conjecture. $\beta_{a,b}^{-s}$ belongs to $\mathcal{G}$ for all $s\geq 1$.

Let $c>1$, and let $U_c$ be the Choquet simplex of probability distributions solving the size-bias scaling relation associated with $c$. Write $c=\exp(\sigma^2/2)$, and let $Z$ be…

Let $A\Gamma_d(\eta,\Sigma)$ be an infinitely divisible positive definite matrix gamma distribution, with total mass $\omega_{d,\eta}$, and consider the limiting spectral distribut…

Marginal infinite divisibility conjecture. The cylindrical probability measure $\mu$ is infinitely divisible if every image measure $\mu\bs a^{-1}$ is infinitely divisible for all…