## Bernoulli

• Bernoulli
• Volume 25, Number 4A (2019), 2824-2853.

### A multivariate Berry–Esseen theorem with explicit constants

Martin Raič

#### Abstract

We provide a Lyapunov type bound in the multivariate central limit theorem for sums of independent, but not necessarily identically distributed random vectors. The error in the normal approximation is estimated for certain classes of sets, which include the class of measurable convex sets. The error bound is stated with explicit constants. The result is proved by means of Stein’s method. In addition, we improve the constant in the bound of the Gaussian perimeter of convex sets.

#### Article information

Source
Bernoulli, Volume 25, Number 4A (2019), 2824-2853.

Dates
Revised: September 2018
First available in Project Euclid: 13 September 2019

https://projecteuclid.org/euclid.bj/1568362044

Digital Object Identifier
doi:10.3150/18-BEJ1072

Mathematical Reviews number (MathSciNet)
MR4003566

Zentralblatt MATH identifier
07110113

#### Citation

Raič, Martin. A multivariate Berry–Esseen theorem with explicit constants. Bernoulli 25 (2019), no. 4A, 2824--2853. doi:10.3150/18-BEJ1072. https://projecteuclid.org/euclid.bj/1568362044

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#### Supplemental materials

• Proofs of certain technical issues. The supplementary file contains a proof of continuous differentiability of $f_{A}^{\varepsilon}$ in Lemma 2.1, and proofs of Lemmas 3.3 and 3.4.