Local increment inference for time-inhomogeneous drift in Gaussian processes

Abstract We study statistical inference for deterministic drifts in Gaussian process models under high-frequency observations over an expanding time horizon. Using a least squares-type contrast based on first-order increments, we establish consistency and asymptotic normality under conditions on drift accumulation and increment dependence. A key feature is that the convergence rate is determined jointly by the deterministic signal and the full covariance structure of the weighted Gaussian increments, rather than by local noise roughness alone. For power and fixed-frequency periodic drifts under Gaussian and Ornstein–Uhlenbeck covariance kernels, we derive explicit convergence rates and limiting variances, revealing distinct regimes depending on the drift structure and, for periodic drifts, the noise spectrum. These results clarify the respective roles of sampling frequency and observation horizon.