Strong approximation of the empirical process of GARCH sequences
type: Item
instance of: scholarly article
title: Strong approximation of the empirical process of GARCH sequences
zbMATH Open document ID: 1014.60033
author: István Berkes
author: Lajos Horváth
published in: The Annals of Applied Probability
publication date: 2003-05-06T00:00:00Z
review text: The authors deal with a GARCH\((p,q)\) process defined by the equations \[ y_k=\sigma_k\varepsilon_k \quad\text{and}\quad \sigma_k^2 =\delta +\sum_{1\leq i\leq p} \beta_i \sigma_{k-i}^2 + \sum_{1\leq j\leq q} \alpha_j y_{k-j}^2 \] where \(\delta\), \(\beta_i\), and \(\alpha_i\) are nonnegative constants and \(\varepsilon_i\) are iid random variables. Let \(F\) be the distribution function of \(y_0\). Define the empirical process \(R(s,t)=\sum_{1\leq i\leq t} (I\{y_i\leq s\} -F(s))\). Under the assumption that there exists a unique stationary solution \(y_k\), a strong approximation for \(R(s,t)\) is obtained. This result implies the weak convergence of the empirical process and the law of the iterated logarithm.
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60G10
zbMATH DE Number: 1906240
zbMATH Keywords: GARCH
zbMATH Keywords: empirical process
zbMATH Keywords: strong approximation
zbMATH Keywords: rates of convergence
zbMATH Keywords: estimates for increments
reviewed by: Jiří Anděl
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1015345349
OpenAlex ID: W1976998709
cites work: Approximation theorems for independent and weakly dependent random vectors
cites work: Q4097679
cites work: Fractionally integrated generalized autoregressive conditional heteroskedasticity
cites work: The sample ACF of a simple bilinear process
cites work: Q4127104
cites work: Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation
cites work: ARCH models and financial applications
cites work: Subadditive ergodic theory
cites work: Reproducing kernel Hilbert spaces and the law of the iterated logarithm for Gaussian processes
cites work: Q3997990
cites work: Q4085018
cites work: Q4327561
cites work: Q3687431
cites work: Almost sure approximation theorems for the multivariate empirical process
cites work: An invariance principle for the law of the iterated logarithm
DOI: 10.1214/AOAP/1015345349
description: scientific article; zbMATH DE number 1906240