31 problems

Stochastic diameter monotonicity conjecture. If

Let $\mu$ be a CNA+ measure on the Boolean lattice $\mathcal{B}_n$, with coordinate indicators $X_1,\ldots,X_n$. Conditional-cover conjecture. The conditional distribution on…

Let $\mu$ be a CNA+ measure, and condition it on the total rank $\sum_eX_e$. Rank-cover conjecture. For each $k$, the conditional distribution given total rank $k+1$ stochastically…

Let $k\in\mathbb{N}$, and let $Z_{X_i}$ denote the random quantity associated with the subpopulation or partition indexed by $i$. For $1\leqslant i\leqslant\lfloor k/2\rfloor$, con…

Let $T_n(f)$ be the preferential attachment tree on $[n]$ generated by the nondecreasing attachment function $f:{\mathbb{N}}_0\to\mathbb{R}_{>0}$. Write $D_n(f)$ for the graph dist…

Let $T$ be an infinite tree whose vertex degrees are all at least $3$, and let a simple symmetric random walk of length $n$ take place on $T$. A regeneration is an edge crossed exa…

Let $\mathcal{A}$ be the system of probability distributions under consideration. Synchronizability conjecture. The synchronizability of $\mathcal{A}$ is somewhat necessary for the…

Let $\lambda_1,\lambda_2:[0,\infty)\to[0,\infty)$ be arrival-rate functions, let $x>0$, let $g\in\mathcal{G}$, and let $\Psi\in\mathcal{L}$. Write $A_\lambda(g;x,\Psi)$ for the ass…

Consider a family of smooth multivariate probability distributions. Multivariate monotonicity conjecture. It should be possible to formulate reasonable conditions under which there…

For two probability measures … , define $\preceq_{\texttt{MCH}}$ when there is a finite sequence of elementary operations transforming … . Let $U_d$ be the sign-vector space, let…

Stochastic height-order conjecture. The stochastic relation on distances to uniformly random leaves also holds for tree heights:

Let $n>1$ and consider the distance from a typical general vehicle to its $n$th-nearest neighbor in the PLM-PPP and in the PSP-PPP, with Rayleigh-distributed half-lengths. Nearest-…

Best-systems conjecture. Regardless of the message-size distribution,

Kochar–Xu conjecture. If $(\lambda_1,\ldots,\lambda_n)\prec(\theta_1,\ldots,\theta_n)$, then

Let $p$ and $q$ be symmetric unimodal weight functions, meaning that $i\mapsto p(i)$ and $i\mapsto q(i)$ are symmetric and decreasing for $i\geq0$. Let $p\preceq_Mq$ denote majoriz…

Let $\kappa_1\leq\kappa_2$ be jump rates, and let $Z_t^{\kappa}$ denote the corresponding partition function in a random environment. Write $\preceq_{cx}$ and $\preceq_{cv}$ for th…

Let $Z^1$ and $Z^2$ be partition functions in the space-time random-environment model, and let $\preceq_M$ denote majorization and $\preceq_{cv}$ the concave order. The relation…

Let $Q_1^b(\infty)$, $Q_1^h(\infty)$ and $Q_1^c(\infty)$ denote the stationary numbers of needy patients in the Erlang-R models with blocking and holding and in the closed ward mod…

Let $W^{\mathrm{NA}}$ and $W^{\mathrm{A}}$ denote the stationary waiting times of the repairman in the non-alternating machine-repair model and of the picker in the two-carousel al…

Let $f$ be a nonzero quadratic form and let $\Sigma$ be a nonzero positive semidefinite matrix. Write $W_{f,\Sigma}$ for the corresponding Wald random variable. Quadratic lower-bou…

Let ${\mathcal A}$ be a set of graphs, let $n$ be a positive integer, let $R_n \in_u {\mathcal A}$ be uniformly distributed over ${\mathcal A}_n$, and let…

Let $\Phi$ be a homogeneous sub-Poisson point process with intensity $a>0$, and let $\Phi_a$ be the homogeneous Poisson point process of the same intensity. Write $r_c(\Phi)$ for t…

Perturbed-lattice monotonicity conjecture. Within this class, the critical radius $r_c$ is monotone with respect to the $dcx$ order on the underlying point processes.

The $dcx$ monotonicity conjecture. If

Let $n\geq 2$, let $A_n=(1,1,\ldots,1)$ be the uniform discount sequence, let $M>0$, and let $F$ be a nondegenerate prior distribution. Write $W(MF,\alpha_2;A_n)$ for the bandit's…