Related papers: A central limit theorem for functions of stationar…

We obtain the law of large numbers (LLN) and the central limit theorem (CLT) for weakly dependent non-stationary arrays of random fields with asymptotically unbounded moments. The weak dependence condition for arrays of random fields is…

We obtain central limit theorems for stationary random fields employing a novel measure of dependence called $\theta$-lex weak dependence. We show that this dependence notion is more general than strong mixing, i.e., it applies to a broader…

This paper investigates a local central limit theorem for a normalized sequence of random variables belonging to a fixed order Wiener chaos and converging to the standard normal distribution. We prove, without imposing any additional…

We establish the central limit theorem for linear processes with dependent innovations including martingales and mixingale type of assumptions as defined in McLeish [Ann. Probab. 5 (1977) 616--621] and motivated by Gordin [Soviet Math.…

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…

Likelihood inference for max-stable random fields is in general impossible because their finite-dimensional probability density functions are unknown or cannot be computed efficiently. The weighted composite likelihood approach that…

We present normal approximation results at the process level for local functionals defined on dynamic Poisson processes in $\mathbb{R}^d$. The dynamics we study here are those of a Markov birth-death process. We prove functional limit…

We study central limit theorems for certain nonlinear sequences of random variables. In particular, we prove the central limit theorems for the bounded conductivity of the random resistor networks on hierarchical lattices.

This is an extended version of a series of talks I held at the University of Bochum in 2017 about limit theorems for non-linear functionals of stationary Gaussian random fields. The goal of these talks was to give a fairly detailed…

We study functional central limit theorems for persistent Betti numbers obtained from networks defined on a Poisson point process. The limit is formed in large volumes of cylindrical shape stretching only in one dimension. The results cover…

We study the asymptotic shape of the trajectory of the stochastic gradient descent algorithm applied to a convex objective function. Under mild regularity assumptions, we prove a functional central limit theorem for the properly rescaled…

We develop goodness-of-fit tests for max-stable random fields, which are used to model heavy-tailed spatial data. The test statistics are constructed based on the Fourier transforms of the indicators of extreme values in the heavy-tailed…

Let $(X_i,i\geq 1)$ be a sequence of i.i.d. random variables with values in $[0,1]$, and $f$ be a function such that $`E(f(X_1)^2)<+\infty$. We show a functional central limit theorem for the process $t\mapsto \sum_{i=1}^n f(X_i)1_{X_i\leq…

We prove central limit theorem under diffusive scaling for the displacement of a random walk on ${\mathbb Z}^d$ in stationary divergence-free random drift field, under the ${\mathcal H}_{-1}$-condition imposed on the drift field. The…

We study the extremes for a class of a symmetric stable random fields with long range dependence. We prove functional extremal theorems both in the space of sup measures and in the space of cadlag functions of several variables. The limits…

We present a central limit theorem for stationary random fields that are short-range dependent and asymptotically independent. As an application, we present a central limit theorem for an infinite family of interacting It\^o-type diffusion…

In this paper, we consider U-statistics whose data is a strictly stationary sequence which can be expressed as a functional of an i.i.d. one. We establish a strong law of large numbers, a bounded law of the iterated logarithms and a central…

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…

The paper investigates isotropic random fields for which the spectral density is unbounded at some frequencies. Limit theorems for weighted functionals of these random fields are established. It is shown that for a wide class of…

We consider "randomized" statistics constructed by using a finite number of observations a random field at randomly chosen points. We generalize the invariance principle (the functional CLT), the Glivenko--Cantelli theorem, the theorem…