10 problems

Persistent hub conjecture. If

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 $\alpha>1$. For the two interacting urn model with power reinforcement $W(n)=n^\alpha$, let … and … where $\mathcal D$ is the domination event. The critical-parameter equality…

Bertoin's recurrence conjecture. The random walk $\bm{S}$ is recurrent. The theorem establishes recurrence in this regime under the stronger moment assumption…

Let $\{W(n)\}_{n\in\mathbb{N}}$ be a strong reinforcement sequence, meaning … and let $\mathcal{M}$ be the event that eventually only balls of one color are added to the urns. Writ…

Let $\{W(n)\}_{n\in\mathbb{N}}$ be a polynomial reinforcement sequence, and let $\mathcal{M}$ be the event that eventually only balls of one color are added to the urns. Write…

Full-support conjecture. This support hypothesis is systematically satisfied for the asymptotic urn variable $\mathbf{W}$.

Conjecture on the nonresonant condition. If there exists $j\geq 2$ such that $q=c\alpha_j$, the proofs should still work after changing the asymptotic expression for a certain quan…

Non-vanishing conjecture. For $N\geq 2$ and $\gamma=1$, $\mathbf{Q}_{n}$ generally does not converge in probability to zero.

Let $(X_n)$ be an exchangeable process, and suppose that $\mathbb{P}(X_{n+1}=i\mid X_0,\ldots,X_n)$ depends only on the number of times $i$ occurs among $X_0,\ldots,X_n$ and on the…