A note on the empirical process of strongly dependent stable random variables

TL;DR

This paper investigates the limit behavior of the empirical process of strongly dependent alpha-stable variables, revealing it converges to a Gaussian process and applying this to goodness-of-fit testing.

Contribution

It introduces a new analysis method for empirical processes of dependent stable variables using Hermite polynomial expansions, extending previous results.

Findings

Limiting process is Gaussian for the empirical process of dependent stable variables.

Provides a weak uniform reduction principle for the empirical process.

Application to goodness-of-fit testing for dependent stable data.

Abstract

This paper analyzes the limit properties of the empirical process of $\alpha$-stable random variables with long range dependence. The $\alpha$-stable random variables are constructed by non-linear 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 $0<\alpha<2$. 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.