Quasi Limiting Distributions on generalized non-local in time and discrete-state stochastic processes

In this article, we study the asymptotic behavior of a discrete space-state and continuous-time killed process \(({\widetilde{X}}^{\nu }(t))_{t \ge 0}\) whose transition probabilities are governed by a non-local convolution type-operator \(\mathcal {D}^{\nu }\). Approximation formulas are provided for small and large values of \(t \ge 0\). In the latter case, the problem of the existence of a Quasi limiting distribution (QLD) is studied in detail, proving that (i) The QLD strongly depends on the initial distribution and (ii) the definition of Quasi Stationary Distribution (QSD) and QLD differs, excepting some very particular cases. Previous to the statement of our main results, a detailed description of our kind of processes is presented. This article generalizes the previous work [25], which is focused on a one-dimensional fractional birth and death process with transition probabilities governed by a fractional Caputo-Dzhrbashyan derivative.