45 problems

Let $\mu$ be a probability measure on the unit circle with infinite support, decomposed as … and let $\alpha=(\alpha_n)$ be its Verblunsky coefficients. For distinct angles…

Let $K\subset\mathbb{R}^{d}$ be a compact set, and let $\mathcal{M}(K)$ be the space of probability measures on $K$ with its weak topology. For a measure $\mu\in\mathcal{M}(K)$, wr…

Let $(\alpha_n)_{n\ge 1}$ be an $L^r$-rate-optimal sequence of $n$-quantizers for a probability measure $P$, and consider the lower bound for its $L^s(P)$-quantization rate of conv…

Let $q,t,a,b\in(0,1)$, let $\rho=(m-1,m-2,\ldots,0)$, and let … For ${\mathbf x}=q^{\lambda}t^{\rho}\in\Omega_m^q$, let $\texttt{dS}_m^q({\mathbf x};a,b,t)$ be the $q$-Selberg meas…

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 $\mu$ be a measure on a Boolean lattice. It is ULC+ when every measure obtained from it by imposing an external field and projecting is ULC. ULC implication conjecture. If…

For a measure on a Boolean lattice, its rank sequence is the sequence of probabilities of the possible values of the total number of occupied coordinates. It is ULC (ultra-log-conc…

Let $\Omega$, $\rho_0$, and $\rho_1$ be as in the preceding convergence theorem, and let $F_{\theta_j}:\Omega\to\Omega$ be a sequence of invertible RealNVP neural networks whose pu…

Let $\mu$ and $\nu$ be compactly supported probability measures, let $c\in[0,\infty]$, and let $\mu\oplus_c\nu$ denote their $c$-convolution. Let $f\in C^{4}(\mathbb{R})$ satisfy…

Limiting-measure and operator conjecture. The family $\hat\nu_\kappa^\sigma$ converges weakly as $\sigma\to0$ to a measure $\hat\nu_\kappa$ on \hat\mathcal A. Moreover, the opera…

Small-order higher-derivative convergence conjecture. For every probability measure $\mu$ on the complex plane and every sequence $k_n$ satisfying $k_n=o(n)$, $\mu_{P_n^{(k_n)}}$ c…

Let $S_n$ be the symmetric group and let $\mathbb F_r$ be the free group on $r$ generators. For a word with constants $w\in S_n\ast\mathbb F_r$, let $\mu_w^{S_n}$ be the distributi…

Orthogonal identifiability conjecture. Under suitable regularity conditions, the conclusion of Proposition remains valid up to an orthogonal transformation for any two Borel probab…

Let $\mathcal{P}_{2,c}(\mathbb{R})$ be the space of centered probability measures on $\mathbb{R}$ with finite second moment, and for $\mu_0,\mu_1\in\mathcal{P}_{2,c}(\mathbb{R})$ d…

Let $S^{n-1}$ be the unit sphere, let $p>0$ be not even, and let ${\operatorname{PFP}}(\mu,p)$ denote the probabilistic $p$-frame potential for a probability measure $\mu$ on…

Let $cPhi_t$ denote the flow map of the cubic Szegő equation on $coperatorname{BMO}_+ = cPi(L^\infty(\mathbb{T}))$, and let $cmu_s$ be the associated measure for $s>\frac12$, so th…

Let $(\Omega,\sigma(\mathscr{H}))$ be the measurable space associated with the sub-linear expectation framework, and let $\varphi:\mathbb R^{\mathbb N}\to\mathbb R$ be a Borel meas…

Dzhafarov–Kujala proportionality conjecture. In the class of cyclic systems, $\text{CNT}_{3}$ and $\text{CNT}_{2}$ should be proportional.

Let $\mathcal{P}$ and $\mathcal{Q}$ be families of probability measures on $\mathfrak{X}$. Suppose there exist $P_0\in\mathcal{P}$ and $Q_0\in\mathcal{Q}$ such that $P\ll P_0$ for…

Let $\Theta$ be an index set, and let $\{P_i,Q_j\}_{i,j\in\Theta}$ be probability measures on $\mathfrak{X}$. Suppose there is some $\theta\in\Theta$ such that $P_i\ll P_\theta$ an…

Let $\sigma$ be a non-degenerate, non-atomic Radon measure on $\mathbb{R}^{d}$ with $\sigma(\mathbb{R}^{d})=\infty$, let $\mathcal{D}=\mathcal{D}(\mathbb{R}^{d})$ be the space of s…

Let $T>0$, let $\mathbb{T}^d$ be the $d$-dimensional torus, let $\mathcal{P}(\mathbb{T}^d)$ denote the space of probability measures on it, and let $(f_N)_{N\geq 1}$ be the regular…

Weak convergence conjecture. On ${\mathcal P}_{\Gamma,q,\ell}$, the measures $nu_{q,\ell,n}$ converge weakly, as $n\to\infty$, to the measure $mu_{\Gamma,q}$ from the existence and…

Existence and uniqueness conjecture. There exists a unique probability measure $mu_{\Gamma,q}$ on ${\mathcal P}_{\Gamma,q}$ such that, for every $(H,h)\in{\mathcal P}_{\Gamma,q}$,

For $gamma in \, ]1,2[$ and $\widetilde{c} in \, ]0,\infty[$, let $\Lambda_{\gamma,\widetilde{c}}$ be the function associated with the corresponding power divergence construction i…