Related papers: Random polynomials: central limit theorems for the…

We consider the number of roots of linear combinations of a system of $n$ orthogonal eigenfunctions of a Sturm-Liouville initial value problem with i.i.d. standard Gaussian coefficients. We prove that its distribution inherits the…

The general model of coagulation is considered. For basic classes of unbounded coagulation kernels the central limit theorem (CLT) is obtained for the fluctuations around the dynamic law of large numbers (LLN). A rather precise rate of…

We consider the asymptotic normalcy of families of random variables $X$ which count the number of occupied sites in some large set. We write $Prob(X=m)=p_mz_0^m/P(z_0)$, where $P(z)$ is the generating function $P(z)=\sum_{j=0}^{N}p_jz^j$…

The critical beta-splitting tree, introduced by Aldous, is a Markov branching phylogenetic tree. Aldous and Pittel recently proved, amongst other results, a central limit theorem for the height of a random leaf. We give an alternative…

We prove a central limit theorem (CLT) for the product of a class of random singular matrices related to a random Hill's equation studied by Adams$\unicode{x2013}$Bloch$\unicode{x2013}$Lagarias. The CLT features an explicit formula for the…

In 1964 Lochs proved a theorem on the number of continued fraction digits of a real number $x$ that can be determined from just knowing its first $n$ decimal digits. In 2001 this result was generalised to a dynamical systems setting by…

We study the number of real critical points of a cyclotomic polynomial $\Phi_{n}(x)$, that is, the real roots of $\Phi_{n}^{\prime}(x)$. As usual, one can, without losing generality, restrict $n$ to be the product of distinct odd primes,…

This paper is concerned with the limiting spectral behaviors of large dimensional Kendall's rank correlation matrices generated by samples with independent and continuous components. We do not require the components to be identically…

The paper deals with a random connection model, a random graph whose vertices are given by a homogeneous Poisson point process on $\mathbb{R}^d$, and edges are independently drawn with probability depending on the locations of the two end…

Let $f_n(z) = \sum_{k = 0}^n \varepsilon_k z^k$ be a random polynomial where $\varepsilon_0,\ldots,\varepsilon_n$ are i.i.d. random variables with $\mathbb{E} \varepsilon_1 = 0$ and $\mathbb{E} \varepsilon_1^2 = 1$. Letting $r_1,…

A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on-or-above diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of…

Let $K$ be a smooth convex set with volume one in $\BBR^d$. Choose $n$ random points in $K$ independently according to the uniform distribution. The convex hull of these points, denoted by $K_n$, is called a {\it random polytope}. We prove…

Let ${\cal T}$ be a rooted Galton-Watson tree with offspring distribution $\{p_k\}$ that has $p_0=0$, mean $m=\sum kp_k>1$ and exponential tails. Consider the $\lambda$-biased random walk $\{X_n\}_{n\geq 0}$ on ${\cal T}$; this is the…

We investigate the probability density of rescaled sums of iterates of deterministic dynamical systems, a problem relevant for many complex physical systems consisting of dependent random variables. A Central Limit Theorem (CLT) is only…

We further explore a connection initially unveiled in Iksanov (2025) between critical beta-splitting trees and infinite `balls-in-boxes' schemes. Using the connection, we derive a new joint central limit theorem for components of the height…

We establish a central limit theorem (CLT) for families of products of $\epsilon$-independent random variables. We utilize graphon limits to encode the evolution of independence and characterize the limiting distribution. Our framework…

We introduce two probabilistic models of random log-concave polynomials, the uniform model and the beta model, and study the asymptotic distribution of their zeros in the complex plane. In the uniform model, we show that the empirical root…

Suppose $B_i:= B(p,r_i)$ are nested balls of radius $r_i$ about a point $p$ in a dynamical system $(T,X,\mu)$. The question of whether $T^i x\in B_i$ infinitely often (i. o.) for $\mu$ a.e.\ $x$ is often called the shrinking target problem.…

We prove a central limit theorem concerning the number of critical points in large cubes of an isotropic Gaussian random function on a Euclidean space.

Motivated by open problems in applied and computational algebraic topology, we establish multivariate normal approximation theorems for three random vectors which arise organically in the study of random clique complexes. These are: (1) the…