In the one-dimensional case Rio (Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 802–817) gave a concise bound for the central limit theorem in the Vaserstein distances, which is a ratio between some higher moments and some powers of the variance. As a corollary, it gives an estimate for the normal approximation of the small jumps of Lévy processes, and Fournier (ESAIM: PS 15 (2011) 233–248) applied that to the Euler approximation of stochastic differential equations driven by the Lévy noise. It will be shown in this article that following Davie’s idea in (Polynomial Perturbations of Normal Distributions. Available at: www.maths.ed.ac.uk/~sandy/polg.pdf (2016)), one can generalise Rio’s result to the multidimensional case, and have higher-order approximation via the perturbed normal distributions, if Cramér’s condition and a slightly stronger moment condition are assumed. Fournier’s result can then be partially recovered.
Mots-clés : Central limit theorem, Lévy processes, stochastic differential equations, approximations
@article{PS_2019__23__112_0,
author = {Zh\={a}ng, X\={i}l{\'\i}ng},
title = {A multi-dimensional central limit bound and its application to the euler approximation for {L\'evy-SDEs}},
journal = {ESAIM: Probability and Statistics},
pages = {112--135},
publisher = {EDP-Sciences},
volume = {23},
year = {2019},
doi = {10.1051/ps/2017021},
mrnumber = {3928265},
zbl = {1411.60092},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2017021/}
}
TY - JOUR AU - Zhāng, Xīlíng TI - A multi-dimensional central limit bound and its application to the euler approximation for Lévy-SDEs JO - ESAIM: Probability and Statistics PY - 2019 SP - 112 EP - 135 VL - 23 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2017021/ DO - 10.1051/ps/2017021 LA - en ID - PS_2019__23__112_0 ER -
%0 Journal Article %A Zhāng, Xīlíng %T A multi-dimensional central limit bound and its application to the euler approximation for Lévy-SDEs %J ESAIM: Probability and Statistics %D 2019 %P 112-135 %V 23 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2017021/ %R 10.1051/ps/2017021 %G en %F PS_2019__23__112_0
Zhāng, Xīlíng. A multi-dimensional central limit bound and its application to the euler approximation for Lévy-SDEs. ESAIM: Probability and Statistics, Tome 23 (2019), pp. 112-135. doi : 10.1051/ps/2017021. http://archive.numdam.org/articles/10.1051/ps/2017021/
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