  In this paper we consider a class of non-local in time telegraph equations.
Recently, it has been proved that the fundamental solutions of such equations
can be interpreted as the probability density function of a stochastic process.
We study the asymptotic behavior of the variance of this process at large and
short times. In this context, we develop a method to construct new examples
such the variance has a slowly growth behavior, extending the earlier results.
Finally, we show that our approach can be adapted to define new
integro-differential operators which are interesting in sub-diffusion
processes.
