18 problems

Let $\mathbf{X}^{(1)},\mathbf{X}^{(2)},\ldots$ be independent and identically distributed $d$-dimensional observations with independent $\operatorname{Exponential}(1)$ coordinates,…

Let $f$ be a smooth stationary Gaussian process, let $K(r)$ denote its correlation function, and let the Poisson approximation refer to the approximation for the locations and size…

Let the conditions of the stability theorem hold, but now for block sizes $n_i$ that are not necessarily equal. Define the mean block size and variance by … If the relative standar…

Let $d=2$ and suppose that Assumption-(i) and the moment condition hold, in particular $\operatorname{\mathbb{P}}[\omega(e)>0]=1$ for $e\in E_d$. For $n\in\mathbb{N}$, set … and ……

Randomly shifted Gumbel convergence conjecture. There exists a random variable $Z>0$ such that

Let $C_{\mathcal{G},\bm{\psi},\bm{\ell},Q}$ be a clustered Archimax copula satisfying the extended max-domain-of-attraction assumption and the standard stable tail dependence funct…

Let $U_1,\ldots,U_N$ be independent random vectors in $\mathbb{R}^d$ with beta density … where $\beta>-1$, $\mathbb B^d=\\{x\in\mathbb{R}^d:\\|x\\|\leq 1\\}$, and $v_m$ denotes the…

Let $T>1$, let $\tau$ be chosen uniformly from $[T,2T]$, and let $0<\alpha<1$ with $\theta\sim(\log\log T)^{-\alpha}$. Consider an interval of size … The smoothing-scale conjecture…

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

Maximum-deviation limit conjecture. There is a random variable $\xi$ such that

Exponential-profile conjecture. The limit

Let $V_N$ be the maximum of the discretized Gaussian free field on the circle with singularity parameter $\alpha\geq 0$. Let $M_{(\tau=1,\alpha,\alpha)}$ denote the critical Morris…

Universal maximum conjecture. The maximum satisfies, for some constant $C$ and a random variable $Z$ called the derivative martingale,

Let $(X_n)_n$ be a cutoff approximation of a centered Gaussian field on $[0,1]^d$ with logarithmic covariance, and set $c_n={\rm Var}(X_n)$. Let $M'$ be the derivative chaos and le…

Typical short-interval maximum conjecture. Let $\epsilon>0$. For $T_1$ sufficiently large depending on $\epsilon$, for all $T_1\leq T\leq 2T_1$, outside a set of “bad” values of…

Fyodorov and Keating's conjecture. Let $\epsilon>0$. For $T_1$ sufficiently large depending on $\epsilon$, for all $T_1\leq T\leq 2T_1$, outside a set of “bad” values of $T$ of mea…

Convergence-rate conjecture.

Maximum-law conjecture. There is a constant $c>0$ and a limiting random variable $G_d$ such that $Y_t$ converges in law to $G_d$ as $t\to\infty$, with