16 problems

Monotonicity conjecture. The function $\Sigma^2(k)$ is monotone increasing in $r$.

For the Zig-Zag Sampler (ZZS), let $\lambda$ denote the refresh rate of the velocity. Zero-refresh-rate conjecture. The choice $\lambda=0$ is optimal in terms of minimising the asy…

Consider a countably infinite state space with $lambda_i>0$ for all $i\geq 0$, $mu_i>0$ for all $i\geq 1$, retention probabilities $q_i^\pm$ defined for all $i\geq 0$ with…

Let $q_\theta(w)$ be a proposal density for the multiplicative noise that may depend on the parameter $\theta$, rather than satisfying the assumption that $q_\theta(w)$ is independ…

Recursive asymptotic-variance conjecture. For each $f\in\mathcal{B}_b(\mathbb{R}^d)$,

Let ${\varmathbb{T}} = {\varmathbb{I}}-\theta{\mathrm{Proj}}$, where ${\mathrm{Proj}}$ denotes the projection onto one of the sets $A$, $B$, or $C$. Then…

GI/G/1 BRAVO conjecture. The BRAVO effect persists under the more relaxed assumptions considered in the source, beyond the finite fourth-moment and busy-period tail assumptions pre…

Edge-density variance positivity conjecture. One should have $\sigma^2_\psi>0$ for this edge-density functional and for every non-zero linear combination of $\psi(H):=1$ and…

Let $\psi$ be a graph functional satisfying the paper's growth condition, let $p_T$ denote the relevant probability associated with a finite tree $T$, and let $\sigma^2_\psi$ be th…

Let $\pi$ be the invariant measure, let $f\in L_0^2(\pi)$, and let $\phi\in L_0^2(\pi)$ be the corresponding solution of the overdamped Poisson equation. Let $\alpha_f$ be the matr…

Let $P$ be a reversible transition matrix with stationary distribution $\pi$, and let its corresponding graph have vertices given by the states and an edge between distinct states…

Let $\pi$ be the invariant target measure of the perturbed underdamped Langevin dynamics, and let $\sigma_f^2$ denote the asymptotic variance of an observable $f$. Assume the frict…

Let $P_s$ and $P_m$ be pseudo-marginal Metropolis--Hastings kernels formed by averaging $s\leq m$ estimators, and let $\mathrm{V}(\varphi,P_k)$ denote the asymptotic variance of an…

Let $\Sigma^2$ be the universal asymptotic variance defined by … Here $\sigma^2(g)$ is McMullen's asymptotic variance and ${\bf P}$ is the Bergman projection. Universal asymptotic…

Variance-condition conjecture. These equivalent conditions should be sufficient for the almost sure conditional central limit theorem.

Consider a nested sampling problem with independent and identically distributed prior components and a factorising likelihood … Let $V_d$ denote the asymptotic variance of the nest…