12 problems

Let $(P_n)_n$ be a renewal covering of the natural numbers as defined in the paper. Let $Z$ be a random variable. Second-order limit conjecture. … where the convergence is in distr…

Maximality conjecture. The polynomial $Q_k$ is a maximal element of $\mathop{\mathtt{A}}_k$ and is the largest element of $\widehat{\mathop{\mathtt{A}}}_k$ for every $k\geq 3$.

Let $S$ be a random walk satisfying assumption, let $O_n$ be its chain of overshoots started from $x$, and let $\pi_+$ denote the stationary distribution of the overshoot chain. Fo…

Let $F$ be a displacement distribution and let $N_0$ be the random counting measure constructed from the renewal and random-walk occupation measures in the paper, with distribution…

Asymptotic decomposition conjecture. As $N\to\infty$,

Critical-threshold conjecture. There is a constant $a_c=a_c(\alpha)$ such that the big-jump regime holds when $$

Let $M(t)$ be the expected number of independent uniformly chosen numbers from $[0,1]$ needed until their sum exceeds $t$, and let $N(t)$ be the expected number of independent unif…

Let $(X_t,\,t\geqslant 0)$ be a linearwise Markov process with state space $\mathscr{X}=\{\mathbb{Z}_+\times\mathbb{R}_+\}$, whose first component records the number of customers i…

Let $S$ be an aperiodic random walk whose increment distribution belongs to $D(\alpha,\rho)$, where $\alpha\in(0,1)$ and $\rho>0$. Let and denote the conditions stated in the sourc…

Let $K_{n,1}$ denote the number of singleton blocks in a regenerative composition of $n$. Assume that the function $t\mapsto t\Phi'(t)$ is nondecreasing, Condition A holds with par…

Cheliotis–den Hollander conjecture. For every $\mu_0$,

Let $F(t)$ be a distribution function on $\mathbf{R}_+=[0,\infty)$ with $F(0)=0$, and let $M(t)$ be the renewal function satisfying … A distribution on $\mathbf{R}_+$ is DFR (decre…