13 problems

Let $d$ be the dimension, let $cmathcal H(d)$ denote the Hankel subspace, and let $p^*=\max\{p,p/(p-1)\}$ be the conjugate exponent. Write $C(\mathcal H(d),p)$ for the optimal…

Let $W_\infty^{(x)}$ and $D_\infty^{(x)}$ denote the truncated martingales started from position $x$, and let $\kappa$ be the parameter appearing in the tail behavior of…

Let $\mathrm{T}\colon S^{d-1}\to\mathbb{R}^{\ell}$ be sufficiently smooth, at least Hölder continuous, and weakly cancelling: … For $\alpha\in(0,d)$, define … let…

Let $\mathcal{N}_t^2$ be the set of type-$2$ particles at time $t$, with positions $X_u(t)$, and let $\overline Z_\infty$ denote the previously defined limit of the relevant multit…

Let $\mathcal{Y}^*=\bigcup_{k\in\mathbb{N}\setminus\{0\}}\mathcal{Y}^k$ be the set of nonempty finite strings over $\mathcal{Y}$, let $\{Y_k:k\in\mathbb{N}\}$ be the Markov chain a…

Let $X$) be a separable Banach space. For integers $N,M\geq 0$, let $(f_{n,m}:0\leq n\leq N,0\leq m\leq M)$ be an $X$-valued martingale on $(\Omega,\mathcal{F},\mathbb{P})$ with re…

Converse characterisation conjecture. If the empirical distributions $(\eta_{n,m})$ form a reverse measure-valued martingale, then $X$ is separately exchangeable, meaning that for…

Let $w\subset\mathbb{R}^d$ be a bounded closed connected set with non-empty interior, and suppose that its boundary is a real-analytic submanifold of $\mathbb{R}^d$. The real-analy…

Let $\Omega_0$ be a compact convex subset of $\mathbb{R}^2$ with non-empty interior. Let $\Omega_1$ be a convex open set such that … Set $\Omega=\Omega_0\setminus\Omega_1$. For a b…

Let $\omega\in 2^{\mathbb{N}\times\mathbb{N}}$. A real is Doob random if it satisfies the paper's characterization in terms of computable martingales, and it is e.c.u. random if it…

Let $A,B esubset \mathbb R+$ be sets of nonnegative real numbers. The set $B$ anticipates $A$ when the regular gamblers' wagers from $B$ can be evaded against gambler 0's wagers…

Downcrossing variance conjecture. For any $\boldsymbol{0}$-feasible process,

Let $(\varepsilon_n)$ be a martingale difference sequence satisfying the homogeneity condition … and assumption $(H_p)$ introduced in the preceding theorem, for some integer…