3 problems

Let $\Omega_1$ be the space of decreasing nonnegative sequences with sum $1$, and let $\widehat{\mu}_1$ denote the Poisson-Dirichlet distribution on $\Omega_1$ with parameter…

Let $\ell_1,\ell_2,\dots$ be the loop lengths ordered decreasingly, let $m(\beta)$ be the long-loop mass, and let $u$ be the model parameter. Define … Write…

Let $\rho\trimmed$ be a stationary random measure on $\mathbb{R}$ with intensity-one asymptotic density, let $\mathscr{K}$ be the disjoint union of…