We consider stationary processes with long memory which are non-Gaussian and represented as Hermite polynomials of a Gaussian process. We focus on the corresponding wavelet coefficients and study the asymptotic behavior of the sum of their squares since this sum is often used for estimating the long-memory parameter. We show that the limit is not Gaussian but can be expressed using the non-Gaussian Rosenblatt process defined as a Wiener-Itô integral of order 2. This happens even if the original process is defined through a Hermite polynomial of order higher than 2.
Keywords: Hermite processes, wavelet coefficients, wiener chaos, self-similar processes, long-range dependence
@article{PS_2014__18__42_0,
author = {Clausel, M. and Roueff, F. and Taqqu, M. S. and Tudor, C.},
title = {Wavelet estimation of the long memory parameter for {Hermite} polynomial of gaussian processes},
journal = {ESAIM: Probability and Statistics},
pages = {42--76},
year = {2014},
publisher = {EDP Sciences},
volume = {18},
doi = {10.1051/ps/2012026},
mrnumber = {3143733},
zbl = {1310.42023},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2012026/}
}
TY - JOUR AU - Clausel, M. AU - Roueff, F. AU - Taqqu, M. S. AU - Tudor, C. TI - Wavelet estimation of the long memory parameter for Hermite polynomial of gaussian processes JO - ESAIM: Probability and Statistics PY - 2014 SP - 42 EP - 76 VL - 18 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/ps/2012026/ DO - 10.1051/ps/2012026 LA - en ID - PS_2014__18__42_0 ER -
%0 Journal Article %A Clausel, M. %A Roueff, F. %A Taqqu, M. S. %A Tudor, C. %T Wavelet estimation of the long memory parameter for Hermite polynomial of gaussian processes %J ESAIM: Probability and Statistics %D 2014 %P 42-76 %V 18 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/ps/2012026/ %R 10.1051/ps/2012026 %G en %F PS_2014__18__42_0
Clausel, M.; Roueff, F.; Taqqu, M. S.; Tudor, C. Wavelet estimation of the long memory parameter for Hermite polynomial of gaussian processes. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 42-76. doi: 10.1051/ps/2012026
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