13 problems

Let $F_1=\ldots=F_A$ be the common birth-rate function for $A$ agents, and let the initial state be … For each $a\geq 2$, let $c(A,a)$ denote the tail-dependence constant appearing…

Consider the unbalanced triangular Pólya urns described in the surrounding discussion, including the class of more typical unbalanced triangular urns for which the almost-sure limi…

Consider a discrete-time triangular Pólya urn satisfying conditions A0–A2, A-5, and A-6–A-7, but not necessarily A-8. The relevant limits may be zero with positive probability, so…

Alternating two-player urn distribution conjecture. The random variable $Y\in[0,1]$ has a density with respect to Lebesgue measure on $[0,1]$, is not Beta-distributed, and has a di…

Let $G$ be a countable graph with uniformly bounded degrees. Let $N_t$ be the system of edge weights initialized at $1$ and evolving according to the continuous-time weakly reinfor…

Consider the convergence-rate bounds for the traditional Pólya urn variables under the assumptions and notation of Corollary, with initial composition $\mathbf{x}_0$ and replacemen…

Stable-limit conjecture. Asymptotically, the empirical distribution $\widetilde \mu$ almost surely is the distribution of a sum of independent copies of $\eta$.

Let $w>0$ and let $(a_k)_{k\geqslant 1}$ define the distribution $\mathrm{UL}(w;(a_k)_{k\geqslant1})$. Let … Set $\pi_k=a_{k+1}\mathbbm{E}Z^{k+1}$ for $k\geqslant0$, and let…

Whisker-forest conjecture. Let $G$ be any graph. There exists $\alpha_G$ such that, for all $\alpha>\alpha_G$:

Convergence to equilibrium. For any WARM with $\alpha>1$, there exists a random vector $\vec{X}=(X_1,\dots,X_n)$, supported on the set of linearly-stable and critical equilibria, s…

Protected-node normality conjecture. For $m\leq26$, every additional eigenvalue of $A$ has real part less than $\frac{1}{2}$. Consequently, the number of protected vertices is asym…

Protected-node non-normality conjecture. For every $m\geq27$, the number of protected nodes is not an exceptional asymptotically normal linear combination and has the same non-norm…

The paper considers local limit theorems for the position $Z_n$ of an infinitely colored Pólya urn scheme, with the precise lattice and distributional hypotheses varying across the…