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A note on the solution to the standard normal Stein equation
Abstract
For z∈ℝ we reconsider the solution fz to the standard normal Stein equation, corresponding to the Kolmogorov distance for distributions on the real line. More precisely, we prove the Lipschitz continuity of the related function x↦ ρz(x):=xfz(x) and explain how this observation may help to remove some unnecessary technical conditions from recent Berry-Esseen bounds in the literature around the Malliavin-Stein method. We further show that the Lipschitz constants of the functions ρz are yet not uniformly bounded in z and discuss consequences of this negative result.
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BibTeX
@misc{dobler2026note,
title = {{A note on the solution to the standard normal Stein equation}},
author = {Christian Döbler},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25992},
eprint = {2610.25992},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25992},
}
RIS
TY - UNPB
TI - A note on the solution to the standard normal Stein equation
AU - Christian Döbler
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25992
UR - https://arxiv.org/abs/2610.25992
AB - For z∈ℝ we reconsider the solution f_z to the standard normal Stein equation, corresponding to the Kolmogorov distance for distributions on the real line. More precisely, we prove the Lipschitz continuity of the related function x↦ ρ_z(x):=xf_z(x) and explain how this observation may help to remove some unnecessary technical conditions from recent Berry-Esseen bounds in the literature around the Malliavin-Stein method. We further show that the Lipschitz constants of the functions ρ_z are yet not uniformly bounded in z and discuss consequences of this negative result.
ER -