Representations of solutions to Stein equations and their derivatives
Abstract
We propose pointwise representations of the Stein solution to a weighted first-order Stein equation associated with a univariate absolutely continuous distribution. The representation extends to all the derivatives of the solution and with respect to any derivative of the test function. The mechanism behind these representations is an array of kernels that satisfy two key identities, yielding a specific calculus for the Stein framework considered here. The representations record every correction term appearing by differentiation, explaining how choosing the Stein kernel as weight has a special role in Taylor and pairing arguments. The calculus of the kernel extends these representations to any order and exhibits natural and explicit conditions to obtain a one term representation of the n-th derivative of the solution with respect to exactly one of the three neighbouring test-function orders of derivatives, n−1,n,n+1. These conditions are simple to ensure by choosing the weight accordingly. The representations yield pointwise envelopes, uniform and weighted Stein factors, and ensure sharpness of the constants. We recover the known sharp results for the Gaussian and the Gamma distribution, improve known factors for the rest of the integrated Pearson family, which is the family under which every correction term cancels. We also study the Subbotin and symmetrized Maxwell families, for which the formulas retain additional pointwise terms. Applications sharpen constants in known distributional approximations and use the higher-order calculus to obtain Edgeworth corrections for the Beta approximation of the Pólya--Eggenberger urn and the quartic Subbotin approximation in the critical Curie--Weiss model.
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BibTeX
@misc{rapin2026representations,
title = {{Representations of solutions to Stein equations and their derivatives}},
author = {Ferdinand Rapin and Yvik Swan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.17881},
eprint = {2609.17881},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17881},
}
RIS
TY - UNPB
TI - Representations of solutions to Stein equations and their derivatives
AU - Ferdinand Rapin
AU - Yvik Swan
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.17881
UR - https://arxiv.org/abs/2609.17881
AB - We propose pointwise representations of the Stein solution to a weighted first-order Stein equation associated with a univariate absolutely continuous distribution. The representation extends to all the derivatives of the solution and with respect to any derivative of the test function. The mechanism behind these representations is an array of kernels that satisfy two key identities, yielding a specific calculus for the Stein framework considered here. The representations record every correction term appearing by differentiation, explaining how choosing the Stein kernel as weight has a special role in Taylor and pairing arguments. The calculus of the kernel extends these representations to any order and exhibits natural and explicit conditions to obtain a one term representation of the n-th derivative of the solution with respect to exactly one of the three neighbouring test-function orders of derivatives, n−1,n,n+1. These conditions are simple to ensure by choosing the weight accordingly. The representations yield pointwise envelopes, uniform and weighted Stein factors, and ensure sharpness of the constants. We recover the known sharp results for the Gaussian and the Gamma distribution, improve known factors for the rest of the integrated Pearson family, which is the family under which every correction term cancels. We also study the Subbotin and symmetrized Maxwell families, for which the formulas retain additional pointwise terms. Applications sharpen constants in known distributional approximations and use the higher-order calculus to obtain Edgeworth corrections for the Beta approximation of the Pólya--Eggenberger urn and the quartic Subbotin approximation in the critical Curie--Weiss model.
ER -