11 problems

Joint-pair Gaussianity conjecture. If $(X_1,X_2)$ is a centered multivariate Gaussian, then $X$ is a Gaussian process.

Exchangeable-distribution extension conjecture. The same decomposition should hold when the component lifetimes have any exchangeable distribution.

Let $\underline{X}$ be the random process and let $\pi$ be a finitary permutation as in Theorem 1.1, whose conclusion concerns the equality of the distributions of $\underline{X}$…

Let $E$ and $D$ be Borel data structures, let $R$ denote the randomization Borel data structure with laws $\boldsymbol{\upsilon}$, and let $\operatorname{SYM}(E)$ denote the exchan…

Let $M=(\Omega,P)$ be a transient Markov chain, and let $X=(X_n)_{n\geq 0}$ be a trajectory of $M$. The partially exchangeable $\sigma$-algebra of $X$ is the exchangeable $\sigma$-…

Let $(X_1,X_2,\ldots)$ be an infinite exchangeable sequence, and let $K_n$ be the number of distinct values among its first $n$ terms. For $m\in\{1,2,\ldots,\infty\}$, let…

Let a sequence have equal zeroth- and first-order frequency limits. Powerlessness conjecture. Our test will be powerless on such a sequence. The conjecture concerns the behavior of…

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

De Finetti's conjecture. A recurrent probability measure on a Cartesian product of a countable state space is a mixture of Markov measures if and only if its associated matrix of s…

Let $(X_n)$ be an exchangeable process, and suppose that $\mathbb{P}(X_{n+1}=i\mid X_0,\ldots,X_n)$ depends only on the number of times $i$ occurs among $X_0,\ldots,X_n$ and on the…

Let $W$ be a kernel, or more generally a function, from ${\mathcal S}^2$ to $[0,1]$, where ${\mathcal S}$ is a probability space. Two points $x_1,x_2\in{\mathcal S}$ are twins if…