Stein Kernels and Normal Approximation for Log-Concave Bilinear Forms
Abstract
Jiang, Lee, and Vempala conjectured that if X,Y∈ℝn are independent isotropic log-concave random vectors, then W2(L(⟨ X,Y⟩),N(0,n)) is bounded by a universal constant. Subject to Theorems 1.2 and 2.5 of arXiv:2607.24164v1, we prove this conjecture and a rectangular bilinear-form extension. For independent isotropic log-concave X∈ℝm, Y∈ℝn, and nonzero B∈ℝm× n, put r4(B)=((Tr(B⊤ B))2)/(Tr((B⊤ B)2)). We construct a nonnegative scalar Stein kernel for X⊤ B Y/|B|F whose squared L2 discrepancy is at most 20/r4(B), and consequently obtain the same bound for squared $2$-Wasserstein distance to N(0,1). The proof develops an exact covariance identity and deficit decomposition for trace observables of moment-map Stein kernels, together with a stability theorem for positive Stein kernels under log-concave approximation. Taking B=In yields W22(L((⟨ X,Y⟩)/√n),N(0,1))≤20/n, which is the Jiang--Lee--Vempala conjecture.
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BibTeX
@misc{liu2026stein,
title = {{Stein Kernels and Normal Approximation for Log-Concave Bilinear Forms}},
author = {Tianle Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.27657},
eprint = {2608.27657},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27657},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stein Kernels and Normal Approximation for Log-Concave Bilinear Forms
AU - Tianle Liu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.27657
UR - https://arxiv.org/abs/2608.27657
AB - Jiang, Lee, and Vempala conjectured that if X,Y∈ℝ^n are independent isotropic log-concave random vectors, then W_2(L(⟨ X,Y⟩),N(0,n)) is bounded by a universal constant. Subject to Theorems 1.2 and 2.5 of arXiv:2607.24164v1, we prove this conjecture and a rectangular bilinear-form extension. For independent isotropic log-concave X∈ℝ^m, Y∈ℝ^n, and nonzero B∈ℝ^m× n, put r_4(B)=((Tr(B^⊤ B))^2)/(Tr((B^⊤ B)^2)). We construct a nonnegative scalar Stein kernel for X^⊤ B Y/|B|_F whose squared L^2 discrepancy is at most 20/r_4(B), and consequently obtain the same bound for squared $2$-Wasserstein distance to N(0,1). The proof develops an exact covariance identity and deficit decomposition for trace observables of moment-map Stein kernels, together with a stability theorem for positive Stein kernels under log-concave approximation. Taking B=I_n yields W_2^2(L((⟨ X,Y⟩)/√n),N(0,1))≤20/n, which is the Jiang--Lee--Vempala conjecture.
ER -