On the empirical process of strongly dependent stable random variables: asymptotic properties, simulation and applications

Abstract

This paper analyzes the limit properties of the empirical process of α-stable random variables with long range dependence. The α-stable random variables are constructed by nonlinear transformations of bivariate sequences of strongly dependent gaussian processes. The approach followed allows an analysis of the empirical process by means of expansions in terms of bivariate Hermite polynomials for the full range 0 < α < 2. A weak uniform reduction principle is provided and it is shown that the limiting process is gaussian. The results of the paper different substantailly from those available for empirical processes obtained by stable moving averages with long memory. An application to goodness-of-fit testing is discussed.

Weak convergence of the sequential empirical copula processes under long-range dependence

“…(3.2) Dehling & Taqqu (1989) studied the asymptotic properties of the one-dimensional empirical process, Taufer (2015), Marinucci (2005) and Buchsteiner (2015) also performed the bivariate and multivariate expansion of one dimensional empirical processes in Hermite polynomials. We now present the limiting distribution of all marginal empirical processes in the following lemma since it is useful results to establish the weak convergence of the empirical copula processes under long-range dependent data.…”

“…For literature on the asymptotic properties of the empirical processes with one dimensional case, see e.g., Taqqu (1975), Dehling & Taqqu (1989) and Leonenko & Sakhno (2001). For the bivariate and multivariate cases, see e.g., Marinucci (2005), Taufer (2015) and Arcones (1994), Bai & Taqqu (2013), Buchsteiner (2015) and Mounirou (2016), respectively. The usual way to derive a limiting distribution of multivariate empirical processes in the context of long-memory is using the multivariate unifrom reduction principle.…”

The function-indexed sequential empirical process under long-range dependence

“…For p = 2 this concept was used by Marinucci (2005) and, if L 1 = L 2 and D 1 = D 2 , by Taufer (2015). However, using this approach we can not model dependency between different components.…”