Asymptotic theory for fractional regression models via Malliavin calculus
type: Item
instance of: scholarly article
title: Asymptotic theory for fractional regression models via Malliavin calculus
zbMATH Open document ID: 1256.60018
author: Solesne Bourguin
author: Ciprian A. Tudor
published in: Journal of Theoretical Probability
publication date: 2012-06-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1004.0680
review text: The authors study the asymptotic behavior as \(n\to\infty\) of the sequences \[ S_n= \sum^{n-1}_{i=0} K(n^\alpha B^{H_1}_i)(B^{H_2}_{i+1}- B^{H_2}_i), \] \[ \langle S\rangle_n= \sum^{n-1}_{i=0} K^2(n^\alpha B^{H_1}_i), \] where \(B^{H_1}\) and \(B^{H_2}\) are two independent fractional Brownian motions with Hurst parameters \(H_1\) and \(H_2\), \(K\) is a nonnegative kernel satisfying \[ \int_{\mathbb{R}} K^2(y)\,dy= 1\quad\text{and}\quad \int_{\mathbb{R}} yK(y)\,dy= 0 \] with (possibly) some additional properties, \(\alpha> 0\). The main results are as follows. The authors prove the limits in distribution \[ \lim_{n\to\infty} n^{\alpha+ H_1-1}\langle S\rangle_n = \int_{\mathbb{R}} K^2(y)\,dy L^{H_1}(1, 0), \] \[ \lim_{n\to\infty} n^{\alpha+ H_1-1} S_n= \int_{\mathbb{R}} K^2(y)\,dy W_{L^{H_1}(1,0)}\;\text{if }\alpha< H_1-1, \] where \(L^{H_1}(1,0)\) is the local time of the functional motion \(B^{H_1}\) and \(W\) a Brownian motion independent from \(B^{H_1}\). Also, they prove a theorem on stable convergence of the sequence \(S_n\). The authors use different mathematical techniques and, in particular, classical Itô integration and elements of Malliavin calculus. Reviewer's remark: In the main results, the parameter \(\alpha\) is not specified. (In Lemma 3, the condition \(\alpha- 4H_2+ H_1+ 2> 0\) is used.) For this reason I found it difficult to understand the results.
Mathematics Subject Classification ID: 60G22
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 91G70
zbMATH DE Number: 6050424
zbMATH Keywords: limit theorems
zbMATH Keywords: fractional Brownian motion
zbMATH Keywords: multiple stochastic integrals
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: regression model
zbMATH Keywords: weak convergence
MaRDI profile type: Publication
OpenAlex ID: W2084568548
arXiv ID: 1004.0680
cites work: Q5680823
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DOI: 10.1007/S10959-010-0302-Y
description: scientific article