20 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…

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…

For a copula whose operator tail density is defined using a diagonal norming matrix, suppose that the entries of the matrix are not all equal. Hidden regular variation conjecture.…

Selection conjecture. Each regularly varying random closed set admits a selection which is regularly varying on a suitably chosen nontrivial ideal.

Let $S$ be a finite state space, let Assumption 0 hold, and consider the quantity … Suppose this quantity is regularly varying with index $\alpha\in(1,2)$. Reverse regular-variatio…

Let $X$ be a positive integer-valued random variable satisfying $\mathbb{P}[X=k]=k^{-\alpha}\ell(k)$, where $\ell$ is slowly varying, and let the PARID-model have parameter…

Limit-process conjecture. The limit process of these scaled processes should be the same as the limit process obtained in the setting studied by Pilipenko and Prykhodko.

Let $\mu_\boxtimes^{\gamma,\sigma}$ be the $\boxtimes$-infinitely divisible measure determined by a finite positive Borel measure $\sigma$ on $[0,+\infty]$ and a real number…

Feasibility conjecture. Estimating $U_{n,\lambda n}$ under such weaker, non-parametric power-law assumptions is possible only if $\lambda$ is small and/or the loss function is suff…

Let $Y$ be a phase-type distributed positive random variable, denoted $Y\sim\operatorname{PH}(\pmb{\pi},\boldsymbol{\bm T})$. A distribution is $\alpha$-regular variation determini…

Let $r$, $p$, and the other coefficients and parameters be as in equation, and let $\mathcal{DS}$ denote the set of decreasing solutions and $\mathcal{NSV}$ the set of solutions wi…

Extension conjecture. These results can be further extended to regularly varying coalescents.

Let $(\xi_n)_{n\geq 1}$ be innovations whose right tail satisfies … for some $r>0$ and some slowly varying function $L$. Let $T_M$ denote the passage time appearing in the precedin…

Let $\boldsymbol X$ be a random vector. Suppose that every convex combination of the components of $\boldsymbol X$ is $2\mathcal{RV}$, meaning that it satisfies a second-order regu…

Let $x$ solve the delay differential equation … and let $y$ solve … Assume that $g(0)=0$, $g'(0)=0$, $g$ is increasing and belongs to $C^1$ on $(0,\delta)$, and is regularly varyin…

Let $f$ be the nonlinearity and, for $\epsilon>0$ and $h>0$, let $S_f(\epsilon,h)$ denote the series governing preservation of the asymptotic behaviour of the finite-difference app…

Regular-variation-only conjecture. The law of the iterated logarithm

Let $({\bf Q}_n)_{n\in\mathbb Z}$ and $(M_n)_{n\in\mathbb Z}$ be the coefficient sequences in the multivariate linear recursion considered in the paper. Grey's conjecture. Using Gr…

Conjecture on extending the main theorem. The main result of the theorem is valid for a wider class of processes and generating functions than those covered by condition (C). The c…

Let $X$ be a random variable whose tail $b[3mP(X\geq x)$ is regularly varying at infinity with index $p<-2$ and satisfies … For a random walk with small negative drift $-a$, let…