Related papers: Random processes and Central Limit Theorem in Beso…
Lower and upper bounds are explored for the uniform (Kolmogorov) and $L^2$-distances between the distributions of weighted sums of dependent summands and the normal law. The results are illustrated for several classes of random variables…
We prove a local central limit theorem (LCLT) for the number of points $N(J)$ in a region $J$ in $\mathbb R^d$ specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of $N(J)$ tends to…
Statistical mechanics describes interaction between particles of a physical system. Particle properties of the system can be modelled with a random field on a lattice and studied at different distance scales using renormalization group…
In this paper, we focus on studying central limit theorems for functionals of some specific stationary random processes. In classical probability theory, it is well-known that for non-linear functionals of stationary Gaussian sequences, we…
Random spatial networks-that is, graphs whose connectivity is governed by geometric proximity-have emerged as fundamental models for systems constrained by an underlying spatial structure. A prototypical example is the random geometric…
A classical fact of the theory of almost periodic functions is the existence of their asymptotic distributions. In probabilistic terms, this means that if $f$ is a Besicovitch almost periodic function and $V$ is a random variable uniformly…
There are many open questions pertaining to the statistical analysis of random objects, which are increasingly encountered. A major challenge is the absence of linear operations in such spaces. A basic statistical task is to quantify…
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…
The objects of our interest are the so-called $A$-permutations, which are permutations whose cycle length lie in a fixed set $A$. They have been extensively studied with respect to the uniform or the Ewens measure. In this paper, we extend…
The question of whether the central limit theorem (CLT) holds for the total number of edges in exponential random graph models (ERGMs) in the subcritical region of parameters has remained an open problem. In this paper, we establish the…
A famous result in renewal theory is the Central Limit Theorem for renewal processes. As in applications usually only observations from a finite time interval are available, a bound on the Kolmogorov distance to the normal distribution is…
In this paper, we study the asymptotic behavior of a fully-coupled slow-fast McKean-Vlasov stochastic system. Using the non-linear Poisson equation on Wasserstein space, we first establish the strong convergence in the averaging principle…
The multivariate central limit theorems (CLT) for the volumes of excursion sets of stationary quasi-associated random fields on $\mathbb{R}^d$ are proved. Special attention is paid to Gaussian and shot noise fields. Formulae for the…
Recent theoretical works have characterized the dynamics of wide shallow neural networks trained via gradient descent in an asymptotic mean-field limit when the width tends towards infinity. At initialization, the random sampling of the…
Self-normalized processes arise naturally in many learning-related tasks. While self-normalized concentration has been extensively studied for scalar-valued processes, there are few results for multidimensional processes outside of the…
We use point processes theory to describe the asymptotic distribution of all upper order statistics for observations collected at renewal times. As a corollary, we obtain limiting theorems for corresponding extremal processes.
We consider large non-Hermitian random matrices $X$ with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having…
Suppose $X = (X_x, x$ in $Z^d)$ is a family of i.i.d. variables in some measurable space, $B_0$ is a bounded set in $R^d$, and for $t > 1$, $H_t$ is a measure on $tB_0$ determined by the restriction of $X$ to lattice sites in or adjacent to…
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…
This paper studies the asymptotic spectral properties of a renormalized sample correlation matrix, including the limiting spectral distribution, the properties of largest eigenvalues, and the central limit theorem for linear spectral…