17 problems

Athreya–Karlin conjecture. For the homogeneous generalized Friedman urn model, $\bm N_n$ is asymptotically normal.

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…

Ordered labeled cutoff-time conjecture. Both chains exhibit cutoff around time

Ordered labeled product-chain conjecture. The chains $\mathbf{X}^{\operatorname{ord,lab}}$ and $\mathbf{X}^{\operatorname{ord,lab}}_{\operatorname{prod}}$ exhibit cutoff around the…

Let $H_0$ denote the first hitting time of $0$ for the one-dimensional Friedman random walk, started after one white draw, with the walk encoded by the difference between the numbe…

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…

Let $\mathbf{Y}_{n}$ and $\mathbf{N}_{n}$ denote the quantities associated with the urn model, and let $\mathbf{H}$ be a homogeneous generating matrix. The asymptotic properties of…

Let $C_N$ be a cycle, let $K\ge3$, and consider the $K$-type competing urn scheme with a nonzero initial configuration. For each type $j$, let balls of type $j$ have independent Po…

Let $C_N$ be a cycle with $N$ vertices, and consider the $K$-type competing urn scheme on $C_N$, where $K\ge3$ and each urn contains balls of at most one type. Single-survivor conj…

Let $C_k$ be the cycle graph of length $k$, and consider the $s$-type urn model on $C_k$, with balls of $s$ colours and each urn monochromatic. At each step, a uniformly chosen bal…

Let $G$ be a finite connected graph, and let $x(n)$ denote the associated urn process. Convergence to a point or interval conjecture. There exists either a point $v(G)$ such that…

Strong reinforcement conjecture. If $s_{\infty}<\infty$, then

Let $(B,W)$ denote the numbers of black and white balls in the Friedman urn process, whose transition probabilities are … First passage statistics for this urn process are much clo…

Partitioned-urn conjecture. The variables $xi_{ij}$ are negatively associated conditional on

Pemantle's conjecture. CNC implies CNA. Pemantle also conjectures that NC+ implies NA+.

Let $(A(t),B(t))$ be the fast continuous-time embedding of the simple harmonic urn, let … and write $\mathbb{E}_n$ for expectation conditional on $A(0)=n$ and $B(0)=0$. Let…