Critical LAN and Score Tests for Mixed Fractional Models under High-Frequency Observation at H=3/4

Abstract: We study the critical boundary $H=3/4$ for two mixed fractional models under high-frequency observation, namely mixed fractional Brownian motion and mixed fractional Ornstein--Uhlenbeck. For different reasons, both the supercritical normalization for $H>3/4$ and the subcritical normalization for $H<3/4$ fail at this value. We identify the exact critical first-order scales, prove the corresponding score central limit theorems, and show that once the explicit linear term in the $H$-score is removed, the resulting $(σ,H)$-block is already non-degenerate. Thus, in contrast with the regime $H>3/4$, the critical point is resolved by a single triangular local reparametrization. Building on this reduction, we derive the critical second-order likelihood expansion and establish LAN for both models with fully explicit leading information constants. Motivated by a testing viewpoint from mathematical finance, we additionally derive score-type tests at the critical boundary. In particular, we formulate one-sided procedures for detecting whether the system enters the supercritical side $H>3/4$. The proposed tests are calibrated under the critical constraint $H=3/4$, use a right-tail rejection region dictated by the LAN drift, and admit fully explicit normalizations at the leading order.