90 problems

Let $T_m(N;\alpha)$ denote the coupon-collection time for the interlacing family of coupon distributions studied in the paper, and suppose the random variable is normalized in the…

Let $K_2(X)$ be the Farey set of elements $a/c$ in the class-number-one imaginary quadratic field with discriminant parameter $D=2$ and $|c|^2<X$, and let … For the normalized elli…

Non-Gaussian limit conjecture. The centered and classically scaled perimeter process has a non-Gaussian distributional limit.

Chatterjee's conjecture. The normalized partial sum should not converge in distribution to $\mathcal{CN}(0,1)$ in either the Rademacher or Steinhaus case.

Gaussian limit conjecture. The Gaussian limit results should continue to hold for

Let a class of probability laws arise as limit laws in a limit theorem, and let a random integral representation mean a representation of its members as laws of random integrals of…

Let $q$ be fixed, let $d>0$, and let $k(\mu,d)$, $J_{k(\mu,d)}^B$, $J_{d/2}^A$, and $\Xi_q$ be as in the preceding corollary. For $\xi,\eta\text{ in }\Xi_q$, write…

Logarithmic-tail conjecture. Equation should hold whenever

Stanley's limiting-distribution conjecture. The limit

Let $w$ be a uniformly random permutation in the symmetric group ${\cal S}_n$, and let $\operatorname{as}_n(w)$ denote the length of its longest alternating subsequence. If…

Darling–Erdős conjecture. With probability one, $0<c_1,c_2<\infty$. By the Kolmogorov zero-one law, $c_1$ and $c_2$ are almost-sure constants. The supplied status evidence says tha…

Regularity-reduction conjecture. The regularity conditions required for this central limit theorem can be reduced to those of Proposition 1 of Mykland and Zhang.

Let $\{a_j\}_{j\geq 1}$ be a sequence of integers, and let … be its Dirichlet series, with parameter $\alpha$ determined by convergence in the half-plane…

Independent-sum asymptotics conjecture. The sum $S_n^{\boldsymbol{\kappa}}$ asymptotically behaves the same way as the sum of independent copies of $\boldsymbol{\kappa}$.

The preceding construction defines processes $X^{\mu^H}$ and establishes convergence to fractional Brownian motion $B_H$. Extension conjecture. This result should remain true in th…

Consider the random location deaths (RLD) process with birth probability $p$, death probability $q=1-p$, and parameter $r$. Let $L_n(a)$ be the number of species alive at time $n$…

Cycle-limit conjecture. Distributional limits for the number of cycles of fixed length $2k$ or $2k+1$ should be Gaussian when $\delta>0$; when $\delta<0$, the limit should be

Oldest-vertex triangle decomposition conjecture. As $n\to\infty$, jointly,

Let $q\in[0,1)$ be the jumping rate, let $U_0$ denote the speed of particle $0$, and write $x=\tfrac{1+u}{2}$ when conditioning on $U_0=u$. Let $\mathcal{C}_0^{n,(q)}$ be the set o…

Let $d\in(1,2)$, and let the total cost of the stable matching be denoted by $C_{n,n}^{\rho}$. Consider any suitable rescaling of this total cost. Stable-limit conjecture. The suit…

Let $U_n$, $U^\circ_n$, $U^{-+}_n$, $U^{+-}_n$, and $U^{--}_n$ be the cyclic and alternating $U$-statistics considered in the paper, with their corresponding normalized variables.…

Let $T_k$ be the passage time, and let $b_\lambda$ and $a_\lambda$ be the centering and scaling sequences for which $(T_k-b_\lambda)/a_\lambda$ converges in distribution as…

Let $X$ and $Y$ be independent random variables, both with continuous distributions, and let $\nu_{n}^{\text{1-dim}}(Y,X)$ denote the one-dimensional estimator defined by ordering…

Finite-moment random-walk conjecture. The theorem asserting that there exists a regular vector $\lambda\in\mathfrak{a}^{++}$ such that $(Z_n(\omega)o)$ is $\lambda$-regular almost…

Let $I_{t}^{(a,b)}$ denote the stochastic integral process studied in the paper, and suppose the model is in one of the divergent cases. Let $a'(\cdot)$ and $b'(\cdot)$ be suitable…