7 problems

For $p,q>0$, let $\beta_{p,q}$ be the beta distribution on $[0,1]$ with density … A probability measure is freely infinitely divisible if it is an $n$-fold free additive convolutio…

Let $X_1,X_2,\dots,X_n\in\mathcal{A}_{sa}$ be a free family of random variables, and let $P$ be a selfadjoint polynomial of degree $2$ in noncommuting variables. Say that $P$ exhib…

Let $P$ be a homogeneous polynomial in noncommuting variables with the property that, whenever $X_1,\dots,X_n$ are free copies of a fixed random variable $X$, the distribution of…

Let $S$ have the centered semicircular distribution with variance parameter $1$, written $S\sim\mathcal{S}(0,1)$. Let $r$ be real. Semicircular power conjectures. The following ass…

Let $X$ have the free Poisson distribution with parameters $p$ and $1$, written $X\sim {\rm fp}(p,1)$. Let $r\in(0,1)$ and $p>0$. Free Poisson power conjecture. The random variable…

Let $\mathbf{t}_q$ denote the t-distribution with parameter $q$. A probability distribution is freely infinitely divisible (FID) if it is infinitely divisible with respect to free…

Let $\bm{\beta}_{p,q}$ and $\bm{\beta}'_{p,q}$ denote the beta and beta prime distributions, respectively, and let $\bm{\gamma}_p$ denote the gamma distribution, with parameters in…