14 problems

Consider a stable first-come-first-served $GI/GI/s$ queue with $s$ identical servers, i.i.d. interarrival times of finite mean $a$, i.i.d. service times of finite mean $b$, mutuall…

Let $X(1),\ldots,X(N)$ be an i.i.d. sample, and let $\widehat H_i$, $i=1,\ldots,M$, be a permutation bootstrap sample of the estimator $\widehat H(j_1,j_2)$. Let the scales $j_1$,…

Let $\mathcal{P}$ be a distribution family on $(0,\infty)$ whose members have power-law tails satisfying $P(X>x)\sim x^{-\alpha}$ for some $\alpha>0$. Let the logarithmic coordinat…

Let $X$ be the covariate, let $\theta(X)$ denote its associated extreme-value parameter, and let $\widehat\beta$ and $\beta^*$ be the XLASSO estimator and target parameter, respect…

Let $\mathbf{A}$ be a random matrix and $\mathbf{B}$ a random vector. Say that both $\mathbf{A}$ and $\mathbf{B}$ have tails lighter than the power-law distribution of index…

Fewest-big-jumps conjecture. For reasonable choices of $(c_n)$, for example $c_n=\gamma n$ with $\gamma\in(0,1)$, one should have approximately

Let $alpha=2$, let $(an){n greater than or equal to 1}$ be a normalising sequence as in the definition of $an$, and assume that the tail $overline{F}$ is intermediate regularly var…

Let $T\geq2$, let $\theta^*$ be the unique minimum of $\mathrm{KL}(\mathrm{q}(\mathbf{x};\theta)\mathbin{\|}\mathrm{p}(\mathbf{x}))$, and define the local Hessian mismatch ratio ……

Let $\pi$ be the invariant probability measure of the rough-flow model, supposing that it exists, and let $\tilde{\pi}$ denote its $X$-marginal. Heavy-tail conjecture. Following nu…

Clustering and star-shaped genealogy conjecture. At a typical time, the majority of the population is close to the leftmost particle, and the genealogy of the population is given b…

Let $M^{(a)}$ be the maximum of a random walk with negative drift parameter $a$, and suppose the increments have Weibull-type right tail … The heavy traffic approximation and the i…

Tails-weight conjecture. A general assessment of the Taylor property in the bilinear process is strongly dependent on its tails weight.

Let $K$ be a cone, let $S(n)$ be a random walk with increments $X$, and let … be its exit time from $K$ when started at $x\in K$. Suppose that $X=R\xi$, where $R$ is a non-negative…

Nonexistence conjecture. There are no harmonic functions for ordered random walks with heavy tails.