Quadratic and $p$-th variation of random signed Takagi--Landsberg bridges

2026-09-18

AI Summary

研究随机带符号的Takagi-Landsberg桥，通过Rademacher混沌的矩界证明了在H=1/2时二次变差为t，且探讨了不同H值下p次变差的行为。

Abstract

We study random signed Takagi--Landsberg bridges with independent Rademacher Faber--Schauder coefficients. At index $H=1/2$, we prove that every fixed deterministic refining sequence of partitions with vanishing mesh yields quadratic variation $t$, almost surely and uniformly in $t\in[0,1]$. For partitions that need not be refining, the mesh condition $o(1/\log n)$ is sufficient, and its order is sharp. Our proofs use an operator representation of the quadratic sums and moment bounds for Rademacher chaos. We also show that asymmetric signs can destroy this invariance. At every index $H\ne1/2$, we construct a deterministic refining sequence along which the critical $p$-power sums, with $p=1/H$, have two distinct finite accumulation points almost surely. At $H=1/4$, uniform thirds grids provide an explicit alternative to the dyadic limit. Thus $H=1/2$ is the unique index in the symmetric family at which critical variation is invariant along every fixed deterministic refining sequence. Nevertheless the variation index equals $1/H$ almost surely across a large sequence of partitions.

Problem

• Quadratic variation
• Takagi-Landsberg bridges
• Rademacher coefficients
• Critical p-power sums

Innovation

• Quadratic Variation
• Takagi-Landsberg Bridges
• Rademacher Chaos
• Critical p-power Sums
• Deterministic Refining Sequence