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 context, we study renewal processes which are stopped at the first event of a further independent renewal process whose inter-arrival time distribution can be defective. If this is the case, the resulting arrival process is of an intermediate nature. For non-defective absorbing times, the resulting arrival process is transient, i.e.\ stopped almost surely. For these processes we derive finite time and asymptotic properties. 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.

Figures (11)

• Figure 1: The plot shows the infinite time limits of (\ref{['Ber-Sib']}) in log scale for a Sibuya process (with index $\mu=0.2$) which is stopped by a Bernoulli process with success probability $p$.
• Figure 2: The figure shows the quantities of (\ref{['BB-infty']}) in log scale. Here a Bernoulli process $N_{II}(t)$ (with success probability $p=0.6$) is stopped by another independent Bernoulli process $N_S(t)$ with success probability $p_S$.
• Figure 3: $\mathbb{E} M(t)$ from (\ref{['IAM_1st_moment']}) for several values of $\mathcal{Q}_S$ and with parameters $q_0=0.3$, $q=0.8$. For large times $\mathbb{E} M(t)$ behaves linearly. In particular, for $\mathcal{Q}_S=1$ (type I limit) the finite asymptotic value $\mathbb{E} M(\infty)=p_0/p=3.5$ is approached.
• Figure 4: The variance (\ref{['variance_defber']}) as a function of time for several choices of $\mathcal{Q}_S$. We have chosen here in all curves $q_0=0.3$ as the parameter of the standard Bernoulli $N_{II}(t)$. The stopping DPB $N_S(t)$ has parameter $q=0.8$. For $\mathcal{Q}_S=1$ (blue curve) the AP $M(t)$ is of type I and the variance has the finite asymptotic value $\mathbb{V}\text{ar}M(\infty) = \frac{2p_0^2q}{p^2}+\frac{p_0}{q} \approx 10.85$. For $\mathcal{Q}_S=0$ we have a type II limit $M(t)=N_{II}(t)$ with linear increase of the variance. For $\mathcal{Q}_S \in (0,1)$ the variance exhibits second-order large time asymptotics corresponding to large fluctuations which are most pronounced in the case $\mathcal{Q}_S=1/2$ (cyan curve).
• Figure 5: Time dependence of $\Lambda_{M,\max}(t)$ (formula (\ref{['Max_Ps']})) for which the fluctuations (variance (\ref{['variance_defber']})) take a maximum. We observe that the asymptotic value $1/2$ is rapidly approached. The other parameters are $q_0=0.3$ and $q=0.8$ as in Fig. \ref{['Var_plot']}.

Theorems & Definitions (26)

• Definition 2.1
• Remark 2.2
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Theorem 3.4
• proof