Statistical Estimations for Non-Ergodic Vasicek Model Driven by Two Types of Gaussian Processes

We study the joint asymptotic distribution of the least squares estimator of the parameter $(\theta,\,\mu)$ for the non-ergodic Vasicek models driven by seven specific Gaussian processes. %The similar result concerning to the non-ergodic Ornstein-Uhlenbeck process is a by-product. To facilitate the proofs, we extract two common hypotheses from the covariance functions of the seven Gaussian processes and develop two types of new inner product formulas for functions of bounded variation in the reproducing kernel Hilbert space of the Gaussian processes. The integration by parts for normalized bounded variation functions is essential to the inner product formulas. We apply the inner product formulas of the seven Gaussian processes to check the set of conditions of Es-Sebaiy, Es.Sebaiy (2021).