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Moment Constants for Brownian Affine Approximation
Abstract
We evaluate moment constants for the times during which Brownian motion admits uniform approximation by an unrestricted affine function or by a line through the origin. The first four moments are rational linear combinations of $1$ and ζ(2j+1)/π2j, 1≤ j≤4; the means are 40ζ(3)/π2−4/3 and 21ζ(3)/π2, respectively. Slope integration and a common contour argument give the unrestricted moments, while time inversion relates the anchored law to the reciprocal of Kingman's absorption time. For both models we also give exponential-integral series for the standardized absolute third central moments, with explicit truncation bounds.
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BibTeX
@misc{chen2026moment,
title = {{Moment Constants for Brownian Affine Approximation}},
author = {Yue Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08068},
eprint = {2609.08068},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08068},
}
RIS
TY - UNPB
TI - Moment Constants for Brownian Affine Approximation
AU - Yue Chen
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08068
UR - https://arxiv.org/abs/2609.08068
AB - We evaluate moment constants for the times during which Brownian motion admits uniform approximation by an unrestricted affine function or by a line through the origin. The first four moments are rational linear combinations of $1$ and ζ(2j+1)/π^2j, 1≤ j≤4; the means are 40ζ(3)/π^2−4/3 and 21ζ(3)/π^2, respectively. Slope integration and a common contour argument give the unrestricted moments, while time inversion relates the anchored law to the reciprocal of Kingman's absorption time. For both models we also give exponential-integral series for the standardized absolute third central moments, with explicit truncation bounds.
ER -