21 problems

Let $\Gamma$ be a finitely generated group, and let $\mu$ be a finitely supported probability measure on $\Gamma$. Suppose that the associated random walk satisfies … where $C_x$ d…

Let $T_n$ be a conditioned Galton--Watson tree with $n$ nodes, whose edges carry independent identically distributed integer-valued increments $\eta_e$ with mean $0$, finite varian…

Local-limit-theorem conjecture. Using a local limit theorem, along with an adaptation of the paper's analyses, could potentially give theoretical support to the procedure of Bernst…

Let $R_{\mathrm{DPV}}(n,\theta)$ denote the parameter appearing in the Diophantine condition $\mathrm{DPV}(n,R)$ for an angle $\theta\in[0,2\pi)$. The paper proves, when…

Unbounded-jump random-walk conjecture. In this setup, the local version of the limit law for $V_n(1)$ is also true, and the random walk defined above is recurrent.

Let $Y_n$ be the branching process with immigration considered in the paper, and let denote the additional moment conditions used for the local limit theorem. The lower-deviation d…

Simplified-model local limit conjecture. For all sufficiently large $n$, for the upper extremal point $M_n^+$ and every integer $k$,

Local limit conjecture. Almost surely,

Extension conjecture. It is conjectured that Bougerol's theorem, and therefore also the local limit theorem on $X$, holds for every non-degenerate probability measure with finite s…

Let $S_n$ be the random walk and let $\tau_x$ denote its exit time as in the preceding asymptotic results. Suppose that $\alpha_n\to\infty$. Uniform extension conjecture. The asymp…

Local limit conjecture. For every partition $clambda cvdash n$, uniformly in $n$ and over all integers $k$,

Local limit conjecture. Uniformly for all $n$ and all integers $k$,

Let $n$ be prime, and let $A_{n,k}$ be the number of 3-term arithmetic progressions in a uniformly random $k$-element subset of $\mathbb{Z}/n\mathbb{Z}$. Fixed-size local limit con…

Let $x_i$, for $1\leq i\leq n$, be independent uniform random variables on $[0,1]$, and define … Here $\bar{A}_n$ is the continuous analogue of the 3-term arithmetic progression co…

Let $n$ be prime and let $A_{n,k}$ be the number of 3-term arithmetic progressions in a uniformly random $k$-element subset of $\mathbb{Z}/n\mathbb{Z}$. Write $\mathbb{E}[A_{n,k}]$…

Let $n$ be prime, let $S$ be a uniformly random subset of $\mathbb{Z}/n\mathbb{Z}$, and let $A_n$ denote the number of 3-term arithmetic progressions in $S$, with mean $\mu_n$ and…

Let $\mu$ be a measure on $(0,1]$ such that … Allow $\mu$ to have atoms and a continuous singular part, and suppose its absolutely continuous part is $g(x)\,dx$, with some…

Let $H(\mathbb{Z})$ be the discrete Heisenberg group, let $S$ be the generating set specified earlier in the source, and let $Q$ be the probability measure supported on $S$. Write…

Local discrete Gaussian conjecture. The number of copies of $H$ in $G(n,p)$ should satisfy a local discrete Gaussian limit law.

The paper considers local limit theorems for the position $Z_n$ of an infinitely colored Pólya urn scheme, with the precise lattice and distributional hypotheses varying across the…

Consider the random walk on a randomly oriented lattice described above, whose first coordinate is related to a random walk in random scenery $Z=\sum_k \xi_{S_k}$. Return-probabili…