A radial Laplace estimate for randomized sums

A radial Laplace estimate for randomized sumsZihuan Xu For a centered isotropic random vector X in R^n, the manuscript bounds a radial Laplace kernel by C(tau+tau^2)n^(-3/2), where tau=n^(-1)Var(||X||^2). Combining this estimate with the angular linear-part inequality of Bobkov, Chistyakov and Goetze gives mean Kolmogorov error at most C Lambda log(en)/n for independent uniform spherical projections, under the all-matrix quadratic-form variance bound Var(X^T A X)<=Lambda||A||_HS^2. For martingale rows with unit conditional variances and uniformly bounded unconditional pth moments, 2<p<4, a bounded coupling preserving the first two conditional moments gives error C M n^(-(p-2)/2)log(en)^(p/4). An application of the Klartag-Sodin independent-coordinate estimate gives a log-free bound for initial mixtures of conditionally independent rows. An independent rare Gaussian-variance construction gives the matching power lower bound. The logarithmic gap for general unit-variance martingale rows and the direction-dependent Gaussian comparison for unequal deterministic conditional variances are stated as specific remaining questions. AI assistance: OpenAI Codex was used to develop and check derivations, inspect source papers, and prepare the text and LaTeX. Author: Zihuan XuAffiliation: School of Mathematical Science, Capital Normal UniversityEmail: 1260503035@cnu.edu.cnORCID: 0009-0002-1510-8754License: CC BY 4.0