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 $\zeta(2j+1)/\pi^{2j}$, $1\le j\le4$; the means are $40\zeta(3)/\pi^2-4/3$ and $21\zeta(3)/\pi^2$, 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.