## Discrete-time arrival processes and defectivity

We consider discrete-time simple counting processes $\bigl(\mathcal{N}(t)\bigr)_{t \in \mathbb{N}_0}$ (also known as discrete-time arrival processes) for which $\mathbf{P}(\mathcal{N}(\infty)<\infty) = \mathcal{Q}$, that is of transient ($\mathcal{Q}=0$), recurrent ($\mathcal{Q}=1$) and intermediate type ($\mathcal{Q} \in (0,1)$). Examples of such processes are discrete-time renewal processes, discrete-time renewal processes with a defective interarrival distributions, externally stopped arrival processes. Some special cases of them are characterized by the presence of generalized difference operators, analogs of convolution-type non-local derivatives, in the governing equations. In this talk we will describe the construction of such arrival processes, highlighting the most important properties and showing some applications to random walks on $\mathbb{Z}^d$ and on the triangular lattice.

Area: CS38 - Subordination and time-changed stochastic processes (Alessandro De Gregorio)

Keywords: Discrete-time renewal processes, defectivity

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