Signed Odd-Root Limits for Intrinsic Fréchet Means on the Circle Under Long-Memory Gaussian Subordination
- 1School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
- 2School of Economics, Beijing Institute of Technology, Beijing 100081, China
Abstract
When the density at the antipode of the intrinsic Fréchet mean reaches a critical level, an interval for the mean of a long-memory circular time series that is calibrated to independent-data scaling is too short, and at Hermite rank two, it also has the wrong shape. The antipode is the cut locus, where squared distance has a kink, and the quadratic coefficient of the population objective vanishes exactly when the density there equals 1/(2π). For Yt=G(Zt), subordinating a stationary Gaussian sequence with covariances decaying at rate α∈(0,1), and contact of order 2r at the antipode, the rescaled estimator converges to a signed (2r+1)-st root. The contact order fixes the root and the Hermite rank of the score fixes what sits under it: a Gaussian at rank one, or a Rosenblatt variable at rank two under strong memory. Because the critical density is a knife-edge equality, we also give the limit along an array approaching it: when the quadratic coefficient vanishes at a rate matched to the long-memory scale of the score, the limit is a one-parameter family joining the odd root to a linear one. In hourly wind directions from fifty-three stations, the antipodal density ranges from 0.05 to 0.74 of the critical level; the records identify no subordination and do not settle the memory, so they locate series relative to the critical level rather than testing the limits.
Related
- A Central Limit Theorem for Intrinsic Fréchet Means on Real Grassmannians
- Lp maximal inequalities for Gaussian subordination with applications to Breuer--Major--Donsker principles
- Gaussian Fluctuations of the Gradient of the Fréchet Function and Nonstandard Asymptotics of Sample Fréchet Means
- Gaussian limits for traces of circular beta-ensembles
- Convergence of Kikuchi matrices to Γ-independent and q-Gaussian limits
BibTeX
@article{yan2026signed,
title = {{Signed Odd-Root Limits for Intrinsic Fréchet Means on the Circle Under Long-Memory Gaussian Subordination}},
author = {Hongxuan Yan and Luoyi Sun},
journal = {Fractal and Fractional},
year = {2026},
month = oct,
doi = {10.3390/fractalfract10100725},
url = {https://www.mdpi.com/2504-3110/10/10/725},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Signed Odd-Root Limits for Intrinsic Fréchet Means on the Circle Under Long-Memory Gaussian Subordination
AU - Hongxuan Yan
AU - Luoyi Sun
PY - 2026
DA - 2026/10/17
JO - Fractal and Fractional
DO - 10.3390/fractalfract10100725
UR - https://www.mdpi.com/2504-3110/10/10/725
AB - When the density at the antipode of the intrinsic Fréchet mean reaches a critical level, an interval for the mean of a long-memory circular time series that is calibrated to independent-data scaling is too short, and at Hermite rank two, it also has the wrong shape. The antipode is the cut locus, where squared distance has a kink, and the quadratic coefficient of the population objective vanishes exactly when the density there equals 1/(2π). For Yt=G(Zt), subordinating a stationary Gaussian sequence with covariances decaying at rate α∈(0,1), and contact of order 2r at the antipode, the rescaled estimator converges to a signed (2r+1)-st root. The contact order fixes the root and the Hermite rank of the score fixes what sits under it: a Gaussian at rank one, or a Rosenblatt variable at rank two under strong memory. Because the critical density is a knife-edge equality, we also give the limit along an array approaching it: when the quadratic coefficient vanishes at a rate matched to the long-memory scale of the score, the limit is a one-parameter family joining the odd root to a linear one. In hourly wind directions from fifty-three stations, the antipodal density ranges from 0.05 to 0.74 of the critical level; the records identify no subordination and do not settle the memory, so they locate series relative to the critical level rather than testing the limits.
SN - 2504-3110
ER -