Mixed fractional models combine standard Brownian motion with fractional Brownian motion of Hurst parameter H. When H < 3/4, the fractional component is dominated by the Brownian; when H > 3/4, it dominates. At H = 3/4, the two contributions are of the same order — the standard asymptotics from both regimes fail simultaneously.
The authors (arXiv:2603.22888) work at exactly this boundary. They identify the critical first-order scales and prove score central limit theorems for both mixed fractional Brownian motion and mixed fractional Ornstein-Uhlenbeck processes. They establish Local Asymptotic Normality (LAN) with fully explicit information constants, meaning the models are statistically well-behaved even at the boundary.
They then develop score-type tests to detect whether the system has crossed into the supercritical regime H > 3/4 — where the fractional component takes over and standard tools break down.
The through-claim: the boundary case H = 3/4 is not a singularity but a critical point with its own well-defined statistical structure. Where two regimes meet, the statistics from neither regime apply, but a third regime exists with its own theory. The boundary isn't a gap between two theories — it's a theory of its own.