Related papers: Central Limit Theorems for Diophantine approximant…
We investigate the number of integer solutions to a multiplicative Diophantine approximation problem and show that the associated counting function converges in distribution to a normal law. Our approach relies on the analysis of…
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…
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 use a method developed by Bj\"orklund and Gorodnik to show a central limit theorem (as $T$ tends to $\infty$) for the counting functions $\# \left( \Lambda \cap \Omega_T \right)$ where $\Lambda$ ranges over the space $Y_{2d}$ of…
We prove central limit theorems for Diophantine approximations with congruence conditions and for inhomogeneous Diophantine approximations following the approach of Bj\"{o}rklund and Gorodnik. The main tools are the cumulant method and…
We consider the problem of counting lattice points contained in domains in $\mathbb{R}^d$ defined by products of linear forms and we show that the normalized discrepancies in these counting problems satisfy non-degenerate Central Limit…
Central limit theorems (CLTs) have a long history in probability and statistics. They play a fundamental role in constructing valid statistical inference procedures. Over the last century, various techniques have been developed in…
In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to recent developments. We describe the connections with mathematical topics such as…
Following the approach of Bj$\ddot{\text{o}}$rklund and Gorodnik, we have considered the discrepancy function for lattice point counting on domains that can be nicely tessellated by the action of a diagonal semigroup. We have shown that…
In this paper, we consider the problem of counting Diophantine inequalities with multiple natural constraints. We prove a very general result in this setting using dynamical techniques. More precisely, we consider the joint asymptotic…
Ordinary differential equations obtained as limits of Markov processes appear in many settings. They may arise by scaling large systems, or by averaging rapidly fluctuating systems, or in systems involving multiple time-scales, by a…
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 prove a functional central limit theorem for subgraph counts in a dynamic version of the random connection model. To establish tightness, we develop a dynamic extension of the cumulant method.
The objective of this study is to investigate the limiting behavior of a subgraph counting process. The subgraph counting process we consider counts the number of subgraphs having a specific shape that exist outside an expanding ball as the…
Linear processes are defined as a discrete-time convolution between a kernel and an infinite sequence of i.i.d. random variables. We modify this convolution by introducing decimation, that is, by stretching time accordingly. We then…
The Central Limit Theorem (CLT) establishes that sufficiently large sequences of independent and identically distributed random variables converge in probability to a normal distribution. This makes the CLT a fundamental building block of…
We study inhomogeneous random graphs with a finite type space. For a natural generalization of the model as a dynamic network-valued process, the paper establishes the following results: (a) Functional central limit theorems for the…
A Central Limit Theorem is proved for linear random fields when sums are taken over finite disjoint union of rectangles. The approach does not rely upon the use of Beveridge Nelson decomposition and the conditions needed are similar to…
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…
We establish effective convergence rates in the Doeblin-Lenstra law, describing the limiting distribution of approximation coefficients arising from continued fraction convergents of a typical real number. More generally, we prove…