A multi-dimensional central limit bound and its application to the euler approximation for Lévy-SDEs

School of Mathematics, The University of Edinburgh, Edinburgh, UK

^* Corresponding author: xiling.zhang@ed.ac.uk

Received: 15 October 2017
Accepted: 30 October 2017

Abstract

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.