Proof of the conjecture about inequality of $1/H$-variations of fractional Brownian motion along the equidistant and the Lebesgue partitions

Abstract

We give an analytic proof of the comparison conjectured by Das, Łochowski, Matsuda and Perkowski for the fractional Brownian level-crossing constant: for standard fractional Brownian motion, $c_H>\mathbb{E}|Z|^{1/H}$ when $0<H<1/2$ and $c_H<\mathbb{E}|Z|^{1/H}$ when $1/2<H<1$, where $Z\sim N(0,1)$. At $H=1/2$, both constants equal one. The main tool is Gaussian comparison under ordering of increment variances. It applies because truncated variation is translation invariant and lattice-submodular, as follows from its minimum-total-variation characterization. A tangent variogram produces Brownian motion with an independent Gaussian drift on the upper side, and a periodic Brownian bridge on the lower side. The former has an explicit stationary reflection rate. For the latter we calculate an integrated cyclic crossing count by first-passage Laplace transforms. Elementary hyperbolic, theta-series, and Gamma inequalities then give strict comparisons on both full parameter ranges. No numerical integration, simulation, interval certificate, or quantitative local expansion at $H=1/2$ is used.