Signed Odd-Root Limits for Intrinsic Fréchet Means on the Circle Under Long-Memory Gaussian Subordination

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.

The authors' abstract, as published at the source. Fractal and Fractional, 2026 · DOI ↗