18 problems

Let $S_{\alpha}$ be a positive $\alpha$-stable random variable with density $g_{\alpha}$, where $\alpha\in(0,1]$. A density is hyperbolically completely monotone (HCM) if it has th…

Let $Q$ and $\alpha$ satisfy … and consider the region to the left of the classic Lévy line, where the current $Q$-Fourier-transform representations do not apply. The source notes…

The parameters are related by $Q=2q-1$, and $\frac{2}{\alpha+1-Q}$ is the scaling rate obtained from the general formula at $k=1$. Fractional nonlinear Fokker–Planck scaling-rate c…

Let $\mathcal{D}^-$ be the class of distributions considered in the paper, and let $\mathcal{S}_C$ denote the class of super-Cauchy distributions. Super-Cauchy characterization con…

Let the $1d$ stochastic heat equation have space-time white noise, and consider its spatial average over boxes whose growth rate depends on time. Stable-limit conjecture. Non-Gauss…

Let $X_1,X_2,\ldots$ be i.i.d. draws from a distribution satisfying the paper's assumptions with associated radial-density exponent $\alpha$, and let the asymptotic results of Theo…

In the random connection hypergraph model, let $gamma$ and $gamma'$ denote the model parameters governing the relevant vertex-mark distributions, and let the triangle count denote…

In the random connection hypergraph model, let $gamma$ and $gamma'$ denote the model parameters governing the relevant vertex-mark distributions, and let the Betti numbers denote t…

Consider a neural network with $D$ layers whose weights are independently distributed according to Stable distributions, and suppose the weight distribution is scaled appropriately…

Let ${\bf X}_{\alpha,\rho}$ be the free stable random variable with stability parameter $\alpha>1$ and positivity parameter $\rho\in[1-1/\alpha,1/\alpha]$. A probability distributi…

Rossberg–Jesiak conjecture. Then

Let $S_{\alpha}$ be a positive $\alpha$-stable random variable, let $g_{\alpha}$ denote its density, and define … Let $G_{\alpha}$ denote the density of $S_{\alpha}^{-\beta}$. A fu…

Cauchy–Schlömilch formula explanation conjecture. We conjecture that these two results follow from the Cauchy–Schlömilch formula.

Stable-limit conjecture. There is an explicit positive exponent $q=q(\zeta)<1/2$ such that, as $N\to\infty$, the random vectors

Let $p_\alpha(t,x)$ be the positive stable kernel, and let ${\rm STP}_n$ and ${\rm SR}_n$ denote strict total positivity and sign-regularity of order $n$, respectively. Positive st…

Uniqueness conjecture. The measures $\sigma^k_\beta$ are the only $k$-symmetric $\boxtimes$-stable distributions.

Let $g_{\alpha}(x)$ denote the one-sided Lévy stable probability density, with rational index $\alpha=l/k$ for positive integers $k>l$. Let $_pF_q$ denote a generalized hypergeomet…

A strictly stable probability measure is a stable law, and a probability measure is convex when it satisfies the paper's convexity property for measures. In finite dimensions, fix…