发表机构
Mathematical Institute, University of Oxford; Department of Mathematics, King’s College London（牛津大学数学研究所; 伦敦国王学院数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于归一化幂变差的非参数方法估计路径粗糙度，在Hölder正则等条件下证明一致性并给出收敛速度，适用于分数布朗运动等，并通过数值实验验证。
AI 中文摘要
我们研究了从离散采样观测中非参数估计路径粗糙度的问题。我们的方法基于路径的归一化幂变差概念。该估计器通过比较粗增量与细增量的局部和来识别变差指数。我们在Hölder正则性、非退化粗临界幂和、细临界幂和的均匀局部界以及两个采样尺度的分离条件下证明了路径一致性，并获得了估计器的收敛速度。既不需要临界变差的收敛性，也不需要有限样本统计量的单调性。该结果适用于在显式二进采样规则下的分数布朗运动，以及系数有界且远离零的Schauder级数。对于带符号的Takagi--Landsberg函数，我们在固定参数区间内获得了估计方程每个根的均匀渐近公式和定量界。我们通过Takagi--Landsberg函数和分数布朗运动的数值实验说明了这些结果。
英文摘要
We study the non-parametric estimation of the roughness of a path from discretely sampled observations. Our approach is based on the concept of normalized power variation of a path. The estimator identifies a variation index by comparing coarse increments with local sums of fine increments. We prove pathwise consistency under Hölder regularity, nondegenerate coarse critical power sums, uniform local bounds on fine critical power sums, and separation of the two sampling scales and obtain a convergence rate for the estimator. Neither convergence of critical variation nor monotonicity of the finite-sample statistic is required. The result applies to fractional Brownian motion under an explicit dyadic sampling rule and to Schauder series with bounded coefficients bounded away from zero. For signed Takagi--Landsberg functions we obtain a uniform asymptotic formula and a quantitative bound for every root of the estimating equation in a fixed parameter interval. We illustrate the results with numerical experiments for Takagi-- Landsberg functions and fractional Brownian motion.
