Replicating a renewal process at random times

Abstract

We replicate a renewal process at random times, which is equivalent to nesting two renewal processes, or considering a renewal process subject to stochastic resetting. We investigate the consequences on the statistical properties of the model of the intricate interplay between the two probability laws governing the distribution of time intervals between renewals, on the one hand, and of time intervals between resettings, on the other hand. In particular, the total number ${\mathcal N}_t$ of renewal events occurring within a specified observation time exhibits a remarkable range of behaviours, depending on the exponents characterising the power-law decays of the two probability distributions. Specifically, ${\mathcal N}_t$ can either grow linearly in time and have relatively negligible fluctuations, or grow subextensively over time while continuing to fluctuate. These behaviours highlight the dominance of the most regular process across all regions of the phase diagram. In the presence of Poissonian resetting, the statistics of ${\mathcal N}_t$ is described by a unique `dressed' renewal process, which is a deformation of the renewal process without resetting. We also discuss the relevance of the present study to first passage under restart and to continuous time random walks subject to stochastic resetting.

Figures (8)

• Figure 1: An example of two nested renewal processes with five cycles of replication of the internal renewal process. Renewal events of the internal process are figured by crosses, replication events (or resettings) due to the external renewal process by dots. The intervals of time between two crosses, ${\boldsymbol{\tau}}_1,{\boldsymbol{\tau}}_2,\dots$, have common probability density $\rho(\tau)$. The intervals of time between two dots, ${\boldsymbol{T}}_1,{\boldsymbol{T}}_2,\dots$, have common probability density $f(T)$. The last interval, $B_t$, represents the backward recurrence time or the age of the external process at time $t$, which indicates the time elapsed since the last replication (or resetting). In this example, the total number ${\mathcal{N}}_t$ of internal renewals (i.e., of crosses) is equal to $6$.
• Figure 2: The four different regions of the phase diagram in the ${\theta}_1$--${\theta}_2$-plane. In regions A and B, the number ${\mathcal{N}}_t$ of internal events (figured by crosses in figure \ref{['fig:nested']}) grows linearly with time and has relatively negligible fluctuations around its mean value. In regions C and D, ${\mathcal{N}}_t$ grows subextensively in time and keeps fluctuating. The notation ${\theta}_{1,2}=\infty$ refers to thin-tailed distributions that possess finite moments of all orders. Poissonian resetting (see section \ref{['sec:PoissonReset']}) lies on the line ${\theta}_2=\infty$.
• Figure 3: Probability density $f_{X_{\theta}}(x)$ of the rescaled random variable $X_{\theta}$ entering (\ref{['eq:Ntscal']}), for several values of the index ${\theta}$ (see legend).
• Figure 4: Quantity ${\zeta}_c$ defined by (\ref{['eq:zcdef']}) and entering the estimates (\ref{['eq:ymoms']}) and (\ref{['eq:ylarge']}), plotted against the ratio ${\alpha}={\theta}_2/{\theta}_1$ for several values of ${\theta}_1$ (see legend).
• Figure 5: Ratio $W=V/V_{\rm max}$, where $V$ is the reduced variance of the scaling variable $Y_{{\theta}_1,{\theta}_2}$ (see (\ref{['eq:vdef']})), and $V_{\rm max}$ is given by (\ref{['eq:vmax']}), plotted against the ratio ${\alpha}={\theta}_2/{\theta}_1$ for several values of ${\theta}_1$ (see legend). Thick black curves: limit expressions (\ref{['eq:w0']}) and (\ref{['eq:w1']}).