  In this manuscript, we analytically and numerically study statistical
properties of an heteroskedastic process based on the celebrated ARCH generator
of random variables whose variance is defined by a memory of
$q_{m}$-exponencial, form ($e_{q_{m}=1}^{x}=e^{x}$). Specifically, we inspect
the self-correlation function of squared random variables as well as the
kurtosis. In addition, by numerical procedures, we infer the stationary
probability density function of both of the heteroskedastic random variables
and the variance, the multiscaling properties, the first-passage times
distribution, and the dependence degree. Finally, we introduce an asymmetric
variance version of the model that enables us to reproduce the so-called
leverage effect in financial markets.
