Related papers: On the Weak Convergence of the Function-Indexed Se…

The purpose of this paper is to prove a weak convergence result for empirical processes indexed in general classes of functions and with an underlying $\alpha$-mixing sequence of random variables. In particular the uniformly boundedness…

We offer an umbrella type result which extends weak convergence of the classical empirical process on the line to that of more general processes indexed by functions of bounded variation. This extension is not contingent on the type of…

Weak convergence of the empirical copula process indexed by a class of functions is established. Two scenarios are considered in which either some smoothness of these functions or smoothness of the underlying copula function is required. A…

In this paper, we study the weak convergence of the integrated periodogram indexed by classes of functions for linear processes with symmetric $\alpha$-stable innovations. Under suitable summability conditions on the series of the Fourier…

Let $(X_j)_{j\geq1}$ be a multivariate long-range dependent Gaussian process. We study the asymptotic behavior of the corresponding sequential empirical process indexed by a class of functions. If some entropy condition is satisfied we have…

A time-varying empirical spectral process indexed by classes of functions is defined for locally stationary time series. We derive weak convergence in a function space, and prove a maximal exponential inequality and a…

Let $(X_k)_{k\geq1}$ be a Gaussian long-range dependent process with $EX_1=0$, $EX_1^2=1$ and covariance function $r(k)=k^{-D}L(k)$. For any measurable function $G$ let $(Y_k)_{k\geq1}=(G(X_k))_{k\geq1}$. We study the asymptotic behaviour…

We consider multivariate copula-based stationary time-series under Gaussian subordination. Observed time series are subordinated to long-range dependent Gaussian processes and characterized by arbitrary marginal copula distributions. First…

Fixed point iterations are a fundamental tool in numerical analysis and scientific computing for the approximation of solutions to nonlinear problems. Their convergence is often established via the Banach fixed point theorem, provided that…

We investigate a particular form of weak convergence of the local empirical process.

In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of bounded functions…

The asymptotic behaviour of empirical measures has been studied extensively. In this paper, we consider empirical measures of given subordinated processes on complete (not necessarily compact) and connected Riemannian manifolds with…

The empirical copula process plays a central role in the asymptotic analysis of many statistical procedures which are based on copulas or ranks. Among other applications, results regarding its weak convergence can be used to develop…

We consider the convergence of empirical processes indexed by functions that depend on an estimated parameter $\eta$ and give several alternative conditions under which the ``estimated parameter'' $\eta_n$ can be replaced by its natural…

This article investigates weak convergence of the sequential $d$-dimensional empirical process under strong mixing. Weak convergence is established for mixing rates $\alpha_n = O(n^{-a})$, where $a>1$, which slightly improves upon existing…

We introduce a maximal inequality for a local empirical process under strongly mixing data. Local empirical processes are defined as the (local) averages $\frac{1}{nh}\sum_{i=1}^n \mathbf{1}\{x - h \leq X_i \leq x+h\}f(Z_i)$, where $f$…

We prove a strong approximation result for the empirical process associated to a stationary sequence of real-valued random variables, under dependence conditions involving only indicators of half lines. This strong approximation result also…

We provide an empirical process theory for locally stationary processes over nonsmooth function classes. An important novelty over other approaches is the use of the flexible functional dependence measure to quantify dependence. A…

We study weak convergence of empirical processes of dependent data $(X_i)_{i\geq0}$, indexed by classes of functions. Our results are especially suitable for data arising from dynamical systems and Markov chains, where the central limit…

(English) This monograph aims at presenting the core weak convergence theory for sequences of random vectors with values in $\mathbb{R}^k$. In some places, a more general formulation in metric spaces is provided. It lays out the necessary…