Asymptotic results for long memory LARCH sequences
type: Item
instance of: scholarly article
title: Asymptotic results for long memory LARCH sequences
zbMATH Open document ID: 1032.62078
author: István Berkes
author: Lajos Horváth
published in: The Annals of Applied Probability
publication date: 2003-11-17T00:00:00Z
review text: Let \((y_n,\sigma_n)\) be a long memory LARCH sequence defined by \[ y_k= \sigma_k\varepsilon_k,\quad \sigma_k= a+\sum^\infty_{i=1} b_i y_{k-i}, \] where \(a\neq 0\), \(\{\varepsilon_i\}\) are i.i.d. random variables with \(E\varepsilon_0= 0\), \(E\varepsilon^2_0= 1\), \(E|\varepsilon_0|^p< \infty\) \((p> 4)\), and \(\sum_n b^2_n<(p- 1)/(3(6p)^3\|\varepsilon_0\|^2_p)\). Denote by \(\{W_\gamma(t), t\geq 0\}\) the fractional Brownian motion with parameter \(\gamma\) \((0<\gamma< 1)\). The authors prove that if \(f\) is a twice continuously differentiable function with \(|f''(x)|\leq C(|x|^\alpha+ 1)\) \((0< \alpha< (p- 4)^2/(2p))\), then \[ N^{-3/2- \beta} \sum^{[Nt]}_{n= 1} (f(\sigma_n)- Ef(\sigma_n))@>{\mathcal D}[0, 1]>>\gamma dW_{(3/2)- \beta}(t), \] and \[ N^{-3/2- \beta} \sum^{[Nt]}_{n= 1} (f(y_n)- Ef(y_n))@>{\mathcal D}[0, 1]>> \gamma_1 dW_{(3/2)- \beta}(t), \] where \(d\) is a positive constant and \(\gamma\), \(\gamma_1\) are some constants depending on \(E(\sigma_0f'(\sigma_0))\) and \(E(y_0 f'(y_0))\), respectively. Moreover, for some new LARCH type sequences associated with \((\sigma_n)\), they formulate a stronger version of the above result.
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 60K99
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 2005172
zbMATH Keywords: LARCH sequences
zbMATH Keywords: long range dependence
zbMATH Keywords: asymptotic distributions
zbMATH Keywords: fractal Brownian motion
reviewed by: Ken-ichi Yoshihara
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1050689598
OpenAlex ID: W2037114052
cites work: Noncentral limit theorems and Appell polynomials
cites work: Strong approximation of the empirical process of GARCH sequences
cites work: Q5560061
cites work: Fractionally integrated generalized autoregressive conditional heteroskedasticity
cites work: The empirical process of some long-range dependent sequences with an application to U-statistics
cites work: Non-central limit theorems for non-linear functional of Gaussian fields
cites work: Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation
cites work: STATIONARY ARCH MODELS: DEPENDENCE STRUCTURE AND CENTRAL LIMIT THEOREM
cites work: A model for long memory conditional heteroscedasticity.
cites work: Q3742423
cites work: Limit theorem for polynomials of a linear process with long-range dependence
cites work: Central limit theorem for the empirical process of a linear sequence with long memory
cites work: Q3911791
cites work: On the asymptotic expansion of the empirical process of long-memory moving averages
cites work: Q4301585
cites work: Q3662382
cites work: Weak convergence to fractional brownian motion and to the rosenblatt process
cites work: Convergence of integrated processes of arbitrary Hermite rank
DOI: 10.1214/AOAP/1050689598
description: scientific article; zbMATH DE number 2005172