On discrete-time arrival processes and related random motions

Abstract

We consider three kinds of discrete-time arrival processes: transient, intermediate and recurrent, characterized by a finite, possibly finite and infinite number of events, respectively. In this framework, we study renewal processes which are externally stopped at an independent stopping time which may be defective or non-defective. For defective stopping time, the resulting arrival process is of intermediate nature. For non-defective stopping time, the resulting arrival process is transient, i.e. stopped almost surely. For these processes we obtain finite time and asymptotic properties. Particular attention is devoted to the class of transient renewal processes, that is, renewal processes with defective interarrival times. Among these, we consider two examples: The "Defective Bernoulli Process" and the "Defective Sibuya Process". We validate some analytical results using Monte Carlo simulations. We apply these results to biased and unbiased random walks on the $d$-dimensional infinite lattice and as a special case on the two-dimensional triangular lattice. We study the spatial propagator of the walker and its large time asymptotics. In particular, we observe the emergence of a superdiffusive (ballistic) behavior in the case of biased walks. For geometrically distributed stopping times, the propagator converges to a stationary non-equilibrium steady state (NESS), which is universal in the sense that it is independent of the stopped process. In dimension one, for both light- and heavy-tailed step distributions, the NESS has an integral representation involving alpha-stable distributions.