36 problems

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…

A one-dimensional stable distribution is a probability distribution on the real line whose characteristic function has the stable-law form. Unimodality conjecture. All one-dimensio…

A hyperbolic random graph places vertices in a hyperbolic space, typically $\mathbb{H}^2$, with edges formed according to probabilities that decay with hyperbolic distance; its deg…

A scale-free Gilbert graph assigns each vertex a random connection radius $R_i$ and connects vertices $X_i$ and $X_j$ if and only if $|X_i-X_j|\leqslant R_i+R_j$. The resulting gra…

Let $\Lambda>1$ and let $\alpha\in(0,2)$. For each $\theta>0$, let $c_\theta$ denote the coefficient defined in Proposition corresponding to the asymptotics of the characteristic f…

Let $f\in\mathcal C_c(\mathbb{R}^d)$, let $S_n(f)$ be the centered, spatially averaged directed-polymer partition function defined by … and let $p^*(\beta)$ denote the relevant mom…

Let $(T_{\gamma},X,\mu)$ be an LSV map of the unit interval with $\gamma\geq 0$, and let $\phi(x)=d(x,x_0)^{-1/\alpha}$ for $x_0\in(0,1]$ and $0<\alpha<1$. Consider the normalized…

Kolmogorov–Spitzer conjecture. The normalized passage times $n^{-1/k}T_n$ could converge to a stable limit law.

Let $\gamma$ be in the normal domain of attraction of an $\alpha$-stable law $\nu$, where $0<\alpha<1$. In particular, $\gamma$ has infinite first moment. Let $F(t)$ denote the res…

Let $\mathbf{A}$ be a non-negative Boltzmann half-pyramid, and let … be its height. Height stable-law conjecture. The random variable $H$ is in the domain of attraction of a stable…

Let $d\geq 2$, and consider a generalized central limit theorem in $\mathbb R^d$ whose limiting law is a symmetric $\alpha$-stable probability measure with $\alpha\in(1,2)$. A rate…

Let $S_n^{(d)}$ be the random walk, let $\alpha$ and $L$ be as in the stable-domain-of-attraction condition … and let $\gamma\in(0,\infty)$. Assume that … Let $\zeta_\alpha$ be the…

Let $Z_t(F)$ be the functional of the derivative Gibbs measure associated with branching Brownian motion, and suppose that $F(x)\sim x^{-\alpha}$ as $x\to0$ for some $\alpha$ satis…

Let $\bar{r}(\epsilon,p,n,\underline{\alpha})$ be the infimum of $r_n(\alpha)$ over $\alpha\in(\underline{\alpha},2]$, and let $\bar{r}$ and $\bar{u}$ denote the universal bounds i…

Total-length fluctuation conjecture. As $n\to\infty$,

Let $\alpha\in(1,2)$, and consider approximation of a $d$-dimensional stable law by the relevant normalized sums. The approximation rate is measured in $L^{1}$ or total variation d…

Stable-law value distribution conjecture.

Let $X_1,X_2,\ldots$ be independent identically distributed random variables with symmetric Pareto tails of index $\alpha\in(0,1]$, and let $S_n=n^{-1/\alpha}\sum_{i=1}^n X_i$. The…

Conjecture on the minimum. As $t\to\infty$, for some constant $C>0$,

Mueller–Munier's conjecture. The random variables

Ricard's Cauchy-domain-of-attraction conjecture. The offspring distribution $u$ falls within the domain of attraction of a Cauchy distribution.

Consider a branching law in the domain of attraction of a $(1+\beta)$-stable law, with generating function … where $L$ is slowly varying at $0$. Let the process be in either the su…

Let $X$ have the stable distribution $\mathbf{S}(\alpha,\beta,\gamma,\delta;1)$, with $-1<p<0$, and let $m^p(\alpha,\beta,\gamma,\delta)=E X_+^p$. Define $\delta^*$ by the unit-sca…

Davydov–Nagaev conjecture. The convergence rate should be

Stable-noise scaling-limit conjecture. The sequence