  We study estimation and prediction of Gaussian processes with covariance
model belonging to the generalized Cauchy (GC) family, under fixed domain
asymptotics. Gaussian processes with this kind of covariance function provide
separate characterization of fractal dimension and long range dependence, an
appealing feature in many physical, biological or geological systems. The
results of the paper are classified into three parts. In the first part, we
characterize the equivalence of two Gaussian measures with GC covariance
function. Then we provide sufficient conditions for the equivalence of two
Gaussian measures with Mat{\'e}rn (MT) and GC covariance functions and two
Gaussian measures with Generalized Wendland (GW) and GC covariance functions.
In the second part, we establish strong consistency and asymptotic distribution
of the maximum likelihood estimator of the microergodic parameter associated to
GC covariance model, under fixed domain asymptotics. The last part focuses on
optimal prediction with GC model and specifically, we give conditions for
asymptotic efficiency prediction and asymptotically correct estimation of mean
square error using a misspecified GC, MT or GW model, under fixed domain
asymptotics. Our findings are illustrated through a simulation study: the first
compares the finite sample behavior of the maximum likelihood estimation of the
microergodic parameter of the GC model with the given asymptotic distribution.
We then compare the finite-sample behavior of the prediction and its associated
mean square error when the true model is GC and the prediction is performed
using the true model and a misspecified GW model.
