Asymptotic properties of self-normalized linear processes with long memory
type: Item
instance of: scholarly article
zbMATH Open document ID: 1263.62113
DOI: 10.1017/S026646661100065X
author: Magda Peligrad
author: Hailin Sang
published in: Econometric Theory
publication date: 2012-06-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1006.1572
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 62F03
Mathematics Subject Classification ID: 60G22
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 6045071
zbMATH Keywords: time series
zbMATH Keywords: heavy tails
zbMATH Keywords: fractional Brownian motion
zbMATH Keywords: unit root testing
MaRDI profile type: Publication
arXiv ID: 1006.1572
cites work: The LIL when X is in the domain of attraction of a Gaussian law
cites work: When is the Student \(t\)-statistic asymptotically standard normal?
cites work: Invariance principles for linear processes with application to isotonic regression
cites work: Asymptotics for linear processes
cites work: Q5512461
cites work: The Invariance Principle for Stationary Processes
cites work: Donsker's theorem for self-normalized partial sums processes
cites work: Limit distributions of Studentized means.
cites work: On linear processes with dependent innovations
cites work: UNIT ROOT TESTING FOR FUNCTIONALS OF LINEAR PROCESSES
cites work: Strong approximation for long memory processes with applications
cites work: Weak convergence to fractional brownian motion and to the rosenblatt process
cites work: The Fractional Unit Root Distribution
cites work: Regularly varying functions
cites work: Self-Normalized Processes
cites work: Time Series Regression with a Unit Root
cites work: Parameter estimation for ARMA models with infinite variance innovations
cites work: Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root
cites work: Limit theorems for self-normalized linear processes
title: Asymptotic properties of self-normalized linear processes with long memory
review text: The authors study the convergence to fractional Brownian motion for long memory time series having independent innovations with infinite second moments. Their study is motivated by economic applications, e.g., the so-called unit root testing problem. Thus the process investigated is NEWLINE\[NEWLINE X_k = \sum_{i=0}^{\infty} a_i \epsilon_{k-i}, \quad k \geq 1, NEWLINE\]NEWLINE where the random variables \(\{\epsilon_n : n \in \mathbb{Z}\}\) are i.i.d., centered and in the domain of attraction of a normal law, that is, \(\ell(x):=\int_{|\epsilon_n| \leq x} \epsilon_n^2 d\mathbb{P}\) is a slowly varying function at \(\infty\). It is assumed that the real coefficients \(\{a_n : n \geq 0\}\) decrease as \(a_n = n^{-\alpha} L(n)\), where \(1/2 < \alpha < 1\), and \(L\) is a slowly varying function at \(\infty\).NEWLINENEWLINEThe main result of the paper is that the process \(W_n(t) := S_{\lfloor nt\rfloor} /B_n\), \((0 \leq t \leq 1)\) weakly converges to the fractional Brownian motion \(W_H\) with Hurst index \(H=3/2 -\alpha\), on the Skorokhod space \(D[0,1]\), where \(S_n = \sum_{i=1}^n X_i\) and \(B_n\) is a normalizing constant defined in the paper. Since \(B_n\) depends on the function \(\ell\) which is not known in general, the authors give a self-normalized version of the theorem: NEWLINE\[NEWLINE \frac{S_{\lfloor nt\rfloor}}{n a_n \left(\sum_{i=1}^n X_i^2\right)^{1/2}} \Rightarrow \frac{\sqrt{c_{\alpha}}}{A} W_H(t) , NEWLINE\]NEWLINE where the constants \(c_\alpha\) and \(A\) are defined in the paper and \(\Rightarrow\) denotes weak convergence.
reviewed by: Tamas Szabados
description: scientific article; zbMATH DE number 6045071