37 problems

Let $F((X_t)_{t\in[0,T]})$ be a functional depending on the whole trajectory of a diffusion, and suppose its Euler approximation has a sharp weak-convergence rate of order $c/n$ or…

Let $c4_N$ denote the distribution of the relevant finite planar triangulations, conditioned to have all vertex degrees bounded by $M$. Weak convergence conjecture. For every…

Let $Y$ solve the stochastic differential equation … where $b$ is a $d\times d$ time-homogeneous volatility matrix, and assume that $b$ satisfies standard technical assumptions. Le…

Conditional non-degeneracy conjecture. For fixed $k\geq 2$, the conditional distribution of $\alpha(\mathcal W_{d+k,d})$ converges weakly as $d\to\infty$ to a limit law not concent…

Rare-spike limit conjecture. The sequence of measures $(\mu_d)_{d\in\mathbb N}$ converges weakly to a non-trivial limit.

Weak-limit conjecture. As $d\to\infty$, the random variables

Let $(\mathcal{T}_n)_{n\in\mathbb{N}}$ be stationary random tessellations in $\mathbb{R}^d$ converging to a stationary random tessellation $\mathcal{T}$, and let…

The absence conjecture. There is no normed linear space where an unbounded sequence converges $d$-weakly.

Asymptotic equicontinuity conjecture. Similar arguments could also be used to verify $$ for the sequential process $\mathbb Z_n$.

Smaller-step-size conjecture. The mismatch between the empirical observation and the theoretical prediction may disappear when much smaller step sizes are used.

Step-size conjecture. The step sizes used in the numerical experiments are not small enough for Theorems and to apply.

Empirical-law conjecture. The empirical law converges almost surely, in the Prokhorov topology, to

Let $t\in\mathbb{N}$, let $\psi\in\mathcal{B}_b(\mathbb{R}^{t+2})$ be continuous, and let $h\in\mathcal{B}_b(\mathbb{R}^{d(t+1)})$. For the step-$n$ particles $X_n^{M,m}$, define ……

For the birth Mallows process, let $J_t=|\{k\leq\binom{n}{2}:T_k\leq t\}|$ be the number of jumps by time $t$. Gaussian Markov-limit conjecture. As $n$ tends to infinity, … where…

Let $A_k\subseteq[0,+\infty)$ be finite sets with $\#A_k=3$ for every $k\geq1$, and let $\nu$ be the corresponding infinite convolution. Write … This condition is known to ensure t…

For each $n$, let $\pi^n$ be the invariant distribution of the system with $n$ servers, and let $\pi$ be the unique distribution satisfying the mean-field fixed-point equation. Inv…

Consider the stochastic Maxwell–Bloch system with $\sigma_1=\sigma_3=1$, the proposed explicit stochastic Poisson integrator, and a test function used to measure weak error. Weak c…

Passeggeri's density conjecture. For every $d\in\mathbb{N}$, the class of quasi-infinitely divisible distributions on $\mathbb{R}^d$ is dense in the class of all distributions on…

Density conjecture. The class of QID distributions on $\mathbb{R}^{d}$ with finite quasi-Lévy measure and zero Gaussian variance is dense in the space of probability distributions…

Let $c$ be a global or finite-time solution of equation with $M_1[c]=\rho$, and let $\mu_c(t)$, $g_\lambda$, and the scaling function be as in Theorem 2. The theorem establishes co…

Let $(X,\mathbb{P})$ be a Lévy process, let $\overline{X}_t$ denote its running maximum, let $q_t^*$ be the distribution associated with the ascending ladder process, let $\nu$ be…

Let $V_N$ be the estimated value-function approximation and let $V_0$ be its population counterpart. The paper establishes a weak convergence result for the smoothed approximation,…

Gaussian-process limit conjecture for the vector of $D_{i,K}$. As $K$ tends to infinity, $\widetilde D_K$ converges weakly to a Gaussian process in the product space…

Conditional geometric-limit conjecture. Conditionally on $X_n\ne0$, the random variable $X_n$ converges weakly to a random variable $Y_\infty$ satisfying

Let $\eta_t$ be the bias voter model on $S$, where $S=\mathbb{T}^d$ or $S=\mathbb{Z}^d$ with $d\geq 1$, started from the product initial distribution $\mu_p$ with $p\in (0,1)$. Wea…