Related papers: Central limit theorem and Diophantine approximatio…

Let $X_1,\dots, X_n,\dots$ be i.i.d.\ $d$-dimensional random vectors with common distribution $F$. Then $S_n = X_1+\dots+X_n$ has distribution $F^n$ (degree is understood in the sense of convolution). Let $$ \rho_{\mathcal{C}_d}(F,G) =…

We derive normal approximation bounds in the Kolmogorov distance for sums of discrete multiple integrals and $U$-statistics made of independent Bernoulli random variables. Such bounds are applied to normal approximation for the renormalized…

The paper establishes a functional version of the Hoeffding combinatorial central limit theorem. First, a pre-limiting Gaussian process approximation is defined, and is shown to be at a distance of the order of the Lyapounov ratio from the…

We demonstrate a simple analytic argument that may be used to bound the Levy concentration function of a sum of independent random variables. The main application is a version of a recent inequality due to Rudelson and Vershynin, and its…

For a measure preserving transformation $T$ of a probability space $(X,\mathcal F,\mu)$ we investigate almost sure and distributional convergence of random variables of the form $$x \to \frac{1}{C_n} \sum_{i_1<n,...,i_d<n}…

We establish the limiting distribution of $\frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{n\le x}\alpha(n)$ where $\alpha$ is a Steinhaus random multiplicative function, answering a question of Harper. The distributional convergence is proved…

We study the counting function of rational approximations with given bounds on the denominator and satisfying the critical Dirichlet exponent on the sphere $S^d$, $d\geq 3$. We give an effective estimate for this counting function, with an…

We study Edgeworth expansions in limit theorems for self-normalized sums. Non-uniform bounds for expansions in the central limit theorem are established while only imposing minimal moment conditions. Within this result, we address the case…

In this paper we characterize all distributional limits of the random quadratic form $T_n =\sum_{1\le u< v\le n} a_{u, v} X_u X_v$, where $((a_{u, v}))_{1\le u,v\le n}$ is a $\{0, 1\}$-valued symmetric matrix with zeros on the diagonal and…

We revisit functional central limit theorems for additive functionals of ergodic Markov diffusion processes. Translated in the language of partial differential equations of evolution, they appear as diffusion limits in the asymptotic…

It is shown that two conjectures put forward in the recent article Iksanov and Kostohryz (2025) are true. Namely, we prove a functional central limit theorem (FCLT) and a law of the iterated logarithm (LIL) for a random Dirichlet series…

We study a random partial covering model on the $(d-1)$-dimensional unit sphere, where $N$ spherical caps are placed independently and uniformly at random, each covering a surface fraction of $1/N$. This model provides a continuous…

Let $ p_n(x) $ be a random polynomial of degree $n$ and $\{Z^{(n)}_j\}_{j=1}^n$ and $\{X^{n, k}_j\}_{j=1}^{n-k}, k<n$, be the zeros of $p_n$ and $p_n^{(k)}$, the $k$th derivative of $p_n$, respectively. We show that if the linear statistics…

In this note we present a new special function that behaves like the error function and we provide an approximated accurate closed form for its CDF in terms of both Chebyshev polynomials of the first kind and the error function. Also, we…

Our goal is to finally settle the persistent problem in Diophantine Approximation of finding best linear approximates. Classical results from the theory of continued fractions provide the solution for the special homogeneous case in the…

We investigate in this paper the distribution of the discrepancy of various lattice counting functions. In particular, we prove that the number of lattice points contained in certain domains defined by products of linear forms satisfies a…

The polynomial $f_{2n}(x)=1+x+\cdots+x^{2n}$ and its minimizer on the real line $x_{2n}=\operatorname{arg\,inf} f_{2n}(x)$ for $n\in\Bbb N$ are studied. Results show that $x_{2n}$ exists, is unique, corresponds to $\partial_x f_{2n}(x)=0$,…

We consider distributional approximation by generalized Dickman distributions, which appear in number theory, perpetuities, logarithmic combinatorial structures and many other areas. We prove bounds in the Kolmogorov distance for the…

We characterize the convergence in distribution to a standard normal law for a sequence of multiple stochastic integrals of a fixed order with variance converging to 1. Some applications are given, in particular to study the limiting…

In this article, we quantify the functional convergence of the rescaled random walk with heavy tails to a stable process.This generalizes the Generalized Central Limit Theorem for stable random variables infinite dimension. We show that…