14 problems

Let $(\Gamma_n)$ be a recurrent Markov chain with origin $o$, let $\mu_\theta$ be the law of the coin-toss record obtained by using bias $\theta\geq 0$ whenever $\Gamma_n=o$ and fa…

Let $(X_k)$ be independent copies of a nondegenerate nonnegative random variable, let $S_n=X_1+\cdots+X_n$, and set $Z_n=\sum_{k=1}^n e^{\imath S_k}$. For $p>0$, write…

Let $P_+$ and $P_-$ denote the probabilities of finding the process in the $+$ and $-$ states, respectively, and let $d707_+$ and $d707_-$ denote the corresponding mean duration ti…

Let $\alpha,\beta\in(0,1]$, and let $X_{\alpha}$ and $X_{\beta}$ be independent Mittag-Leffler random variables with power indices $\alpha$ and $\beta$, respectively, and equal tim…

Uniqueness conjecture. These generalized Arcsine distributions are the only fixed points of the involution $\mu\mapsto\nu$.

Conditioned renewal-process FCLT conjecture. As $n\to\infty$, the appropriately scaled conditioned renewal process should satisfy a counterpart to the functional central limit theo…

Let $\tau$ and $\hat\tau$ be renewal sets with tail exponents … , respectively, and let $h_c(\beta)$ and $h_c^a(\beta)$ denote the quenched and annealed critical points of the pinn…

Let $C_k^{1,1}$ be the renewal-sequence random variable whose point probabilities are under consideration, and let $\zeta(j)$ denote the Riemann zeta function. For each $k\geq 1$ a…

Let $\nu$ be a probability measure on $\mathbb R$ with finite mean $m<\infty$, and let $Q_\nu$ be the stationary renewal process on $\mathbb R$ with renewal distribution $\nu$. A w…

Let $\gamma=0$, and let the renewal law satisfy $K(n):=\mathrm{P}(\tau_1=n)\sim L(n)n^{-\rho}$ with $\rho>1$. Let $\mathrm{P}_{{\beta}_N,h,N}^{{\omega}}$ denote the corresponding p…

Let $\tau$ be the renewal process and let $N$ be the system size, so that $\tau/N$ denotes its rescaled renewal set. In the regime $\gamma\geq 1$, the entropy optimization yields t…

Let a cyclic polling system have general renewal arrival processes and deterministic switch-over times. The system is considered in the regime where the deterministic switch-over t…

Let $(Y_{\alpha,\alpha}(u))_{u>0}$ be the process whose one-dimensional distributions are standard exponential, so that $Y_{\alpha,\alpha}(u)\stackrel{\mathrm{d}}{=}R$ for every…

The A5 conjecture. If the tail condition holds with $\alpha=1$, then