Local increment inference for time-inhomogeneous drift in Gaussian processes

Abstract

We study statistical inference for deterministic drift structures in Gaussian process models under high-frequency observations.The observed process consists of a centered stationary Gaussian component combined with a broad class of time-inhomogeneous deterministic drifts. To estimate the drift parameter, we introduce a least squares-type contrast based on first-order increments. We establish consistency and asymptotic normality under weak dependence conditions on the Gaussian component. A central feature of the framework is that the rate of convergence of the estimator depends jointly on the local roughness of the Gaussian noise and the long-time information accumulation structure generated by the drift. The theory accommodates a wide range of drift families, including integrable, polynomial-type, and periodic structures. In particular, different drift densities produce qualitatively different statistical regimes, including non-standard rates of convergence and accelerated rates for persistent or growing deterministic structures.