Spectral estimation for non-linear long range dependent discrete time trawl processes

Paul Doukhan,François Roueff,Joseph Rynkiewicz

doi:10.1214/20-ejs1742

Paul Doukhan, François Roueff + Show 1 more

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https://doi.org/10.1214/20-ejs1742

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Abstract

Discrete time trawl processes constitute a large class of time series parameterized by a trawl sequence (a j) j∈N and defined though a sequence of independent and identically distributed (i.i.d.) copies of a continuous time process (γ(t)) t∈R called the seed process. They provide a general framework for modeling linear or non-linear long range dependent time series. We investigate the spectral estimation, either pointwise or broadband, of long range dependent discrete-time trawl processes. The difficulty arising from the variety of seed processes and of trawl sequences is twofold. First, the spectral density may take different forms, often including smooth additive correction terms. Second, trawl processes with similar spectral densities may exhibit very different statistical behaviors. We prove the consistency of our estimators under very general conditions and we show that a wide class of trawl processes satisfy them. This is done in particular by introducing a weighted weak dependence index that can be of independent interest. The broadband spectral estimator includes an estimator of the long memory parameter. We complete this work with numerical experiments to evaluate the finite sample size performance of this estimator for various integer valued discrete time trawl processes.

Highlights

This behavior is often referred to as X being long range dependent with long memory parameter d∗ = 1 − α∗/2
A very interesting feature of trawl processes is that under the fairly general assumption (1.4) on the seed process, the low frequency behavior of the spectral density is mainly driven by the trawl sequence
In the case of a Levy seed for instance, a Brownian seed process leads to an invariance principle with fractional Brownian motion limit, with Hurst parameter (3 − α∗)/2, and a Poisson seed process leads to an invariance principle with Levy α∗-stable limit, see [4, Theorems 1 and 2]

Summary

Introduction

This behavior is often referred to as X being long range dependent with long memory parameter d∗ = 1 − α∗/2. The definition of negative long memory (d∗ < 0) is generally relying on the behavior of the spectral density at the origin (in particular imposing this spectral density to vanish there). A very interesting feature of trawl processes is that under the fairly general assumption (1.4) on the seed process, the low frequency behavior of the spectral density is mainly driven by the trawl sequence. Xn. Deriving general results applying to a wide class of long range dependent trawl processes raise two major difficulties. The spectral density has a closed form only in particular cases for the seed process and the trawl sequence.

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