Some probabilistic properties and time-changed versions of a renewal process based on Mittag-Leffler waiting times

In this paper, we obtain some additional probabilistic properties of the renewal process $\{\hat{N}_{\alpha}(t)\}_{t\ge0}$, $0<\alpha\le 1$ introduced by Beghin and Orsingher (2010). A time-changed relationship connecting $\{\hat{N}_{\alpha}(t)\}_{t\ge0}$ with its special case $\{\hat{N}(t)\}_{t\ge0}$ by means of the random time process $\{T_{2\alpha}(t)\}_{t>0}$ whose distribution is related to a fractional diffusion equation is established. We compute its various distributional properties such as the variance, factorial moments, moment generating function, moments, covariance in the Laplace domain, etc. We show that the ratios given by $\{\hat{N}_{\alpha}(t)\}_{ t \ge 0}$ and its power over their means tend to $1$ in probability. Moreover, we derive an integral form of its bivariate distribution and describe the scaling limits of its marginal distributions. It is also shown that its one-dimensional distributions are not infinitely divisible. Furthermore, we study the compound version of $\{\hat{N}_{\alpha}(t)\}_{ t \ge 0}$ and discuss an application to ruin theory. Later, we consider two time-changed versions of $\{\hat{N}_{\alpha}(t)\}_{ t \ge 0}$ which are obtained by time-changing it with an independent L\'evy subordinator and its inverse. Some distributional properties and examples are discussed for these time-changed processes.