  The process $(G_t)_{t\in[0,T]}$ is referred to as a fractional Gaussian
process if the first-order partial derivative of the difference between its
covariance function and that of the fractional Brownian motion
$(B^H_t)_{t\in[0,T ]}$ is a normalized bounded variation function. We quantify
the relation between the associated reproducing kernel Hilbert space of $(G)$
and that of $(B^H)$. Seven types of Gaussian processes with non-stationary
increments in the literature belong to it. In the context of applications, we
demonstrate that the Gladyshev's theorem holds for this process, and we provide
Berry-Ess\'{e}en upper bounds associated with the statistical estimations of
the ergodic fractional Ornstein-Uhlenbeck process driven by it. The second
application partially builds upon the idea introduced in \cite{BBES 23}, where
they assume that $(G)$ has stationary increments. Additionally, we briefly
discuss a variant of this process where the covariance structure is not
entirely linked to that of the fractional Brownian motion.
