220 problems

Aldous's conjecture concerns a diffusion on continuum trees whose finite-dimensional mass partitions evolve as Wright–Fisher diffusions with negative mutation rates. Aldous's diffu…

Let $\mathscr{E}_1,\dots,\mathscr{E}_n$ be independent exponential random variables, and consider weighted sums of their centered versions,…

Laplace-transform criterion. The function $L_m$ is the Laplace transform of some random variable if and only if $m=0$, corresponding to a trivial random variable, or $m\ge e$.

Let $Y,Y_1,Y_2,\ldots$ be positive, nondegenerate, independent and identically distributed random variables with distribution function $G$, let $X,X_1,X_2,\ldots$ be nondegenerate…

Strong dualizability conjecture. Every measurable equivalence relation on a standard Borel space is strongly dualizable.

Let $k\ge2$, let $\lambda$ be sufficiently large that the mode is nonzero, and let $m_k(\lambda)$ denote a mode of the Poisson distribution of order $k$, with mean parameter…

Let $f_t$ denote the density function associated with the parameter $t$, for $0\leq t\leq 1$. A density is unimodal if it is nondecreasing up to a mode and nonincreasing thereafter…

LLL mixing and density conjecture. (i) The sequence $(|\mu_{k(i)}|)_{i=1,2,\ldots}$ is strongly mixing as a stochastic process. (ii) Each $|\mu_{k(i)}|$ is contained in a compact s…

Gneiting's conjecture. Any characteristic function with an absolutely continuous component has the substitution property.

Let a class of probability laws arise as limit laws in a limit theorem, and let a random integral representation mean a representation of its members as laws of random integrals of…

Let unitary invariance and functional invariance be principles governing the quantum-mechanical proof of Born's law, and let Gleason's conclusion mean that the relevant probability…

Universality conjecture. Every compound Poisson measure can appear as a limiting distribution of an inhomogeneous INAR(1) process.

Let $Z=(Z_t)_{t\geq 0}$ be the stable process considered in the paper, let $T$ be the associated stopping time, and let $\alpha$ be its stability index. For every $k>0$, the moment…

Let $(X,r,\mu)$ be a Polish metric measure space, meaning that $(X,r)$ is Polish and $\mu$ is a probability measure on its Borel sets. Gromov's $\underline{\square}_1$-metric is ob…

The p-adic Lévy–Khintchine conjecture. $p$-adic fields have the Lévy–Khintchine property.

Let $h$ be a martingale, and let $A_1$ and $A_2$ be operators on martingales. Suppose that $A_i$ and $h$ belong to some subsets of operators and martingales, respectively, which ma…

Let $X_1,X_2,\ldots$ be identically distributed random variables, let $S_n$ denote their normalized sum, and let $D(S_n)$ be the squared Mallows-distance quantity used in the paper…

Let $X_1$ and $X_2$ be independent and identically distributed random variables, and let ${\mathcal{C}}(X)$ denote the Poincaré-type constant defined in the paper. Consider the nor…

Let $(\Gamma_n)$ be a recurrent Markov chain with origin $o$, let $\mu_\theta$ be the law of the coin-toss record obtained by using bias $\theta\geq 0$ whenever $\Gamma_n=o$ and fa…

Let $X$ have the exponential distribution, specifically $X\sim\operatorname{Exp}(1)$ as in the surrounding discussion, and consider Variance symmetrization resistance. Exponential…

Let $X$ have the exponential distribution on $\a0\mathbb{R}_+$, and let $V$- and $h$-symmetrization resistance be considered in the space $\mathcal{P}_c(\mathbb{R})$ of compactly s…

Let $S_N$ be the sum of the discrete random variables considered in the system, and let $\mathbb{E}S_N$ denote its expectation. The coefficient matrix of is the matrix determining…

Rastegar and Roitershtein formulate a characterization theorem with a condition numbered (1.3), also numbered (3) in the preprint, and consider the range $n\ge3$. Rastegar–Roitersh…

Infinite divisibility conjecture. The random variable ${\hat T}_\theta$ is infinitely divisible for every

Let $\mathscr{E}_1,\mathscr{E}_2,\dots$ be independent exponential random variables, let $G\sim N(0,1)$, and let $p_0$ be the crossover parameter used in the paper. Consider all fi…