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

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