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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/ps/2017021/
[1] and , Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Probab. 38 (2001) 482–493. | DOI | MR | Zbl
[2] and , Normal Approximation and Asymptotic Expansions. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons (1976). | MR
[3] , Berry-Esseen bounds and Edgeworth expansions in the central limit theorem for transport distances. Probab. Theory Relat. Fields 170 (2017) 229–262. | DOI | MR | Zbl
[4] , and , Rate of convergence and Edgeworth-type expansion in the entropic central limit theorem. Ann. Probab. 41 (2013) 2479–2512. | DOI | MR | Zbl
[5] , Rates in the Central Limit Theorem and Diffusion Approximation via Stein’s Method. Preprint (2016). | arXiv
[6] , Pathwise Approximation of Stochastic Differential Equations Using Coupling. Available at: www.maths.ed.ac.uk/~sandy/coum.pdf (2014).
[7] , Polynomial Perturbations of Normal Distributions. Available: www.maths.ed.ac.uk/~sandy/polg.pdf (2016).
[8] , Extensions of results of Komlós, Major, and Tusnády to the multivariate case. J. Multivar. Anal. 28 (1989) 20–68. | DOI | MR | Zbl
[9] , Simulation and approximation of Lévy-driven stochastic differential equations. ESAIM: PS 15 (2011) 233–248. | DOI | Numdam | MR | Zbl
[10] , Asymptotic of grazing collisions for the spatially homogeneous Boltzmann equations for soft and Coulomb potentials. Stoch. Process. Appl. 123 (2013) 3987–4039. | DOI | MR | Zbl
[11] and , Rates of approximation in the multidimensional invariance principle for sums of i.i.d. random vectors with finite moments. J. Math. Sci. 167 (2010) 495–500. | DOI | MR | Zbl
[12] , The Euler scheme for Lévy driven stochastic differential equations: limit theorems. Ann. Probab. 32 (2004) 1830–1872. | DOI | MR | Zbl
[13] and , Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 (1998) 267–307. | DOI | MR | Zbl
[14] and , The Euler scheme for SDEs driven by semimartingales, in Stochastic Analysis on Infinite Dimensional Spaces (1994) 141–151. | MR | Zbl
[15] , and , An approximation of partial sums of independent RV’-s, and the sample of DF.I. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 32 (1975) 111–131. | DOI | MR | Zbl
[16] , Sums of Independent Random Variables. Springer-Verlag (1975). | MR | Zbl
[17] , Upper bounds for minimal distances in the central limit theorem. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 802–817. | Numdam | MR | Zbl
[18] , Asymptotic constants for minimal distances in the central limit theorem. Electron. Commun. Probab. 16 (2011) 96–103. | MR | Zbl
[19] , Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6 (1996) 587–600. | DOI | MR | Zbl
[20] , Topics in Optimal Transportation. Vol. 58 of Graduate Studies in Mathematics. American Mathematical Society (2003). | MR | Zbl
[21] , Multidimensional version of the results of Komlós, Major and Tusnády for vectors with finite exponential moments. ESAIM: PS 2 (1998) 41–108. | DOI | Numdam | MR | Zbl
[22] , Estimates for the strong approximation in multidimensional central limit theorem, in Proceedings of the International Congress of Mathematicians III, Beijing (2002) 107–116. | MR | Zbl
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