A General Asymptotic Normality Theorem for Switching Degree Inference under Weak Dependence — Direction-Dependent Long-Run Variance Correction, Moving Block Bootstrap Consistency, and Preservation of Structural Invariance

The inference theory for the Switching Degree established by Tang (2026ag) assumes independent and identically distributed noise. In this paper, we extend the Switching Degree inference to the weak-dependence setting under α-mixing noise and establish three main results. First, the weak-dependence asymptotic normality theorem (general case): the Switching Degree estimator remains √n-asymptotically normal, with its asymptotic variance jointly determined by the instantaneous variance σ² and the long-run variance σ²_LR, V_true = [ σ² + (σ²_LR − σ²) · (δᵀμ_z)² / ‖δ‖_Σ² ] / [ γ(1−γ) · ‖θ_ref‖_Σ² ], where δ = Δθ and μ_z = E[z]. When μ_z = 0 or δᵀμ_z = 0, this formula reduces to the asymptotic variance under independent noise, σ² / [γ(1−γ) · ‖θ_ref‖_Σ²], in which case weak dependence has no effect on the asymptotic variance of the Switching Degree estimator. Second, a precise statement of moving block bootstrap (MBB) consistency: we explicitly formulate conditions and conclusions for MBB consistency, while leaving the rigorous proof as open problem SMT-T36. Third, a structural invariance preservation theorem: the invariance of the Switching Degree under invertible Σ-isometries does not depend on the noise independence assumption. Numerical simulations confirm the necessity of the weak-dependence correction: confidence intervals that ignore weak dependence exhibit severely low coverage, with coverage dropping as low as 73%, while intervals based on the corrected variance or HAC standard errors improve substantially. In finite samples, all methods have coverage slightly below the nominal level (corrected theoretical intervals range from 0.865 to 0.925), attributable to normal approximation bias caused by the nonlinearity of the Switching Degree; this is posed as an open problem. Together with Tang (2026af, 2026ag, 2026ah, 2026ai, 2026aj), this paper completes the statistical inference foundation of Switch Measure Theory. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently directed by the author. DeepSeek assisted with code implementation, data presentation, and text drafting. The author takes full academic responsibility for the final content.