Related papers: On central limit theorems in the random connection…

Let $G$ be an $N \times N$ real matrix whose entries are independent identically distributed standard normal random variables $G_{ij} \sim \mathcal{N}(0,1)$. The eigenvalues of such matrices are known to form a two-component system…

Consider the random graph $G({\mathcal P}_{n},r)$ whose vertex set ${\mathcal P}_{n}$ is a Poisson point process of intensity $n$ on $(- \frac{1}{2}, \frac{1}{2}]^d$, $d \geq 2$. Any two vertices $X_i,X_j \in {\mathcal P}_{n}$ are connected…

We show how a central limit theorem for Poisson model random polygons implies a central limit theorem for uniform model random polygons. To prove this implication, it suffices to show that in the two models, the variables in question have…

The random connection model is a random graph whose vertices are given by the points of a Poisson process and whose edges are obtained by randomly connecting pairs of Poisson points in a position dependent but independent way. We study…

Let $G$ be a group with a non-elementary action on a proper CAT(0) space $X$, and let $\mu$ be a measure on $G$ such that the random walk $(Z_n)_n$ generated by $\mu$ has finite second moment on $X$. Let $o$ be a basepoint in $X$, and…

For each $n$, let $X_n \in \{0,\ldots,n\}$ be a random variable with mean $\mu_n$, standard deviation $\sigma_n$, and let \[ P_n(z) = \sum_{k=0}^n \mathbb{P}( X_n = k) z^k ,\] be its probability generating function. We show that if none of…

Consider the geometric graph on $n$ independent uniform random points in a connected compact region $A$ of ${\bf R}^d, d \geq 2$, with $C^2$ boundary, or in the unit square, with distance parameter $r_n$. Let $K_n$ be the number of…

A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (X_n)_{n\geq 1} is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (G_n)_{n\geq 0}, if it…

Let I_1,...,I_n be independent but not necessarily identically distributed Bernoulli random variables, and let X_n=\sum_{j=1}^nI_j. For \nu in a bounded region, a local central limit theorem expansion of P(X_n=EX_n+\nu) is developed to any…

We study the convergence in distribution norms in the Central Limit Theorem for non identical distributed random variables that is $$ \varepsilon_{n}(f):={\mathbb{E}}\Big(f\Big(\frac 1{\sqrt…

This paper establishes a central limit theorem and an invariance principle for a wide class of stationary random fields under natural and easily verifiable conditions. More precisely, we deal with random fields of the form $X_k =…

Let M be a noncompact metric space in which every closed ball is compact, and let G be a semigroup of Lipschitz mappings of M. Denote by (Y_n)_{n\geq1} a sequence of independent G-valued, identically distributed random variables (r.v.'s),…

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…

We study the central limit theorem in the non-normal domain of attraction to symmetric $\alpha$-stable laws for $0<\alpha\leq2$. We show that for i.i.d. random variables $X_i$, the convergence rate in $L^\infty$ of both the densities and…

A central limit theorem for arrays of symmetric row-wise exchangeable random variables is presented. The result is valid for finite and infinite extendable and non-extendable sequences. Unlike most reported versions of the central limit…

The number of faces of the convex hull of $n$ independent and identically distributed random points chosen on the boundary of a smooth convex body in $\mathbb{R}^d$ is investigated. In dimensions two and three the number of $k$-faces is…

This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for…

We prove a central limit theorem for random sums of the form $\sum_{i=1}^{N_n} X_i$, where $\{X_i\}_{i \geq 1}$ is a stationary $m-$dependent process and $N_n$ is a random index independent of $\{X_i\}_{i\geq 1}$. Our proof is a…

We prove a central limit theorem for a certain class of functions on sparse rank-one inhomogeneous random graphs endowed with additional i.i.d. edge and vertex weights. Our proof of the central limit theorem uses a perturbative form of…

The Central Limit Theorem (CLT) is one of the most fundamental results in statistics. It states that the standardized sample mean of a sequence of $n$ mutually independent and identically distributed random variables with finite first and…